IRLF 


AN 


ELEMENTARY  TREATISE 


ON 


THEORETICAL  MECHANICS' 


BY 


ALEXANDER   ZIWET 

ASSISTANT  PROFESSOR  OF  MATHEMATICS   IN  THE 
UNIVERSITY  OF  MICHIGAN 


IN  THREE  PARTS 


Nefo  gorfc 
THE    MACMILLAN    COMPANY 

LONDON:   MACMILLAN  &  CO.,  LTD. 
IQOI 

All  rights  reserved 


HALLID1E 


COPYRIGHT,  1894, 
BY  MACMILLAN  AND  CO. 


First  printed  November,  1894. 
Reprinted  March,  1901. 


f 

Nortoooti  $reBB 

J.  S.  Gushing  &  Co.  -  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 

THE  present  work  owes  its  existence  mainly  to  the  difficulty  of 
finding  a  good  modern  text-book  suited  to  the  requirements  of  the 
American  student. 

In  England  it  is  customary  to  take  a  thorough  course  in  elementary 
mechanics  (comprising  plane  statics  and  kinetics  of  a  particle)  before 
entering  upon  the  study  of  higher  mathematics ;  and  there  is  no 
lack  of  works  of  this  character  (Loney,  Macgfegor,  Selby,  Thomson 
and  Tait's  Elements,  Hicks,  Robinson,  Browne,  Blaikie,  Parkinson, 
Wormell,  Lodge,  Laverty,  etc.),  some  of  which  are  very  well  adapted 
to  the  purpose.  A  good  course  in  analytic  geometry  and  the  differ- 
ential and  integral  calculus  will  then  prepare  the  student  for  reading 
the  more  advanced  English  works  on  analytical  statics  (Todhunter, 
Minchin,  Routh)  and  rigid  dynamics  (Williamson  and  Tarleton,  Routh, 
Thomson  and  Tait,  Price,  Besant,  etc.).  A  similar  arrangement  is 
presupposed  by  most  of  the  French  and  German  treatises. 

In  many  American  colleges  and  universities,  however,  the  student 
takes  up  the  study  of  mechanics  at  a  later  stage,  after  having  acquired 
a  knowledge  of  the  elements   of  higher  mathematics.     A  somewhat 
different  treatment  of  the  subject  of  mechanics   is   required   in  this* 
case. 

The  present  volume,  which  is  devoted  to  kinematics,  forms  the  first 
of  three  parts  of  nearly  equal  extent.  The  second  part,  after  an  intro- 
duction to  dynamics  in  general,  takes  up  statics ;  it  will  appear  in  the 
fall  of  this  year.  The  third  part,  which  will  be  ready  in  the  fall  of 
1894,  is  devoted  to  kinetics. 

While  the  work  is  intended,  first  of  all,  as  an  introduction  to  the 

v 

102928 


vi  PREFACE. 

science  of  theoretical  mechanics  as  such,  the  author  has  constantly 
kept  in  mind  the  particular  wants  of  engineering  students,  aiming  to 
make  it  serve  as  a  preparation  for  the  practical  applications  of  this 
science,  and  to  bring  out  the  utility  and  importance  of  the  purely 
mathematical  training.  General  theories  are  illustrated  by  special  prob- 
lems and  applications  in  the  text,  and  sets  of  exercises  are  inserted 
to  be  worked  out  by  the  student. 

To  keep  the  whole  work  within  reasonable  bounds,  the  more  ad- 
vanced parts  of  the  subject  had  to  be  strictly  excluded.  Bibliographi- 
cal references  have  therefore  been  given  for  the  use  of  any  who  are 
desirous  to  pursue  the  subject  farther.  In  accordance  with  the  ele- 
mentary character  of  the  work,  these  references  are  not  to  original 
memoirs,  but  to  such  standard  treatises  as  can  be  expected  to  be 
found  in  a  well- assorted  college  library. 

At  a  first  reading,  the  Articles  57-87,  181-214,  221-244,  272-3°5> 
can  be  omitted,  also  some  of  the  applications  and  the  more  difficult 
exercises. 

ALEXANDER  ZIWET. 

ANN  ARBOR,  MICH., 
July,  1893. 


CONTENTS. 


PAGE 

INTRODUCTION i 


CHAPTER  I. 
GEOMETRY  OF  MOTION. 

I.  LINEAR  MOTION;  TRANSLATION  AND  ROTATION         .  .  3 

II.  PLANE  MOTION     .       ..-•.'      .        .        .        .        *  .  .       8 

III.  SPHERICAL  MOTION      .        ,        .        .        .        .        ,  .  .      18 

IV.  SCREW  MOTION T-  .  .19 

V.  COMPOSITION  AND  RESOLUTION  OF  DISPLACEMENTS: 

1 .  Translations ;  vectors    .         .         .         .         ;        »  .  .       24 

2.  Rotations ;   rotors 30 

3.  Screw  motions  or  twists         .        .        .        .        .  .  .      39 

CHAPTER   II. 

KINEMATICS. 

I.  TIME      .        .       .       ;       .       .       .       .  -  ..v.*     .  ...  .      42 

II.  LINEAR  KINEMATICS: 

1.  Uniform  rectilinear  motion ;  velocity     .         ,         .'  .  .       45 

2.  Variable  rectilinear  motion  ;  acceleration    ...  .  -49 

3.  Applications 53 

4.  Rotation ;  angular  velocity ;  angular  acceleration  .  .      66 

III.  PLANE  KINEMATICS: 

i.   Velocity;  composition  of  velocities  ;  relative  velocity  .  .       70 

a.   Applications 74 


PAGE 

70 


viii  CONTENTS. 

III.  PLANE  KINEMATICS.  —  Continued. 

3.  Acceleration  in  curvilinear  motion          .... 

4.  Applications  •••«.. 

5.  Velocities  in  the  rigid  body   .        .        .....     ,3 

6.  Applications  .  .        .        .  !^ 

7.  Accelerations  in  the  rigid  body      .  .        .  II;I 

IV.  SOLID  KINEMATICS: 

1.  Motion  of  a  point  in  a  twisted  curve      ...  T6i 

2.  Velocities  in  the  rigid  body  .         ......  Z64 

3.  Accelerations  in  the  rigid  body      .....  !67 
ANSWERS  .        .        .        . 


OF  THE 

UNIVE 


THEORETICAL   MECHANICS 


INTRODUCTION. 

1.  The  science  of  theoretical  mechanics  has  for  its  object 
the  mathematical  study  of  motion. 

2.  The  idea  of  motion   is  intimately  related  to  the  funda- 
mental ideas  of  space,  time,  and  mass.     It  will  be  convenient 
to  introduce  these  consecutively.     Thus  we  shall  begin  with  a 
purely  geometrical  study  of  motion,  without  regard  to  the  time 
consumed  in  the  motion  and  to  the  mass  of  the  thing  moved, 
the  moving   object    being  considered    as   a   mere   geometrical 
configuration.     This  introductory  branch  of  mechanics  may  be 
called  the  geometry  of  motion. 

3.  The  introduction  of  the  idea  of  time  will  then  lead  us  to 
study  the  velocity   and  acceleration  of  geometrical  configura- 
tions.    This  constitutes  the  subject-matter  of  Kinematics  proper. 
The  name  Kinematics  is,  however,  used  by  many  authors  in  a 
less  restricted  sense,  so  as  to  include  the  geometry  of  motion. 

4.  Finally,  endowing  our  geometrical  points,  lines,  and  other 
configurations  with  mass,  we  are  led  to  the  ideas  of  momentum, 
force,  energy,  etc.     This  part  of  our  subject,  the  most  compre- 
hensive of  all,  has  been  called  Dynamics,  owing  to  the  importance 
of  the  idea  of  force  in  its  investigation.     For  the  sake  of  con- 
venience it  is   usually  divided  into  two  branches,  Statics  and 

PART   I — I  I 


2  INTRODUCTION.  [4. 

Kinetics.  In  statics  those  cases  are  considered  in  which  no 
change  of  motion  is  produced  by  the  acting  forces,  or,  as  it  is 
commonly  expressed,  in  which  the  forces  are  in  equilibrium. 
The  investigations  of  statics  are  therefore  independent  of  the 
element  of  time.  Kinetics  treats  in  the  most  general  way  of 
the  changes  of  motion  produced  by  forces. 


;.]  LINEAR  MOTION. 


CHAPTER   I. 

GEOMETRY   OF  MOTION. 

I.     Linear  Motion ;    Translation  and  Rotation. 

5.  Motion  consists  in  change  of  position. 

6.  We  begin  with  the  simple  case  of  a  point  moving  in  a 
straight  line.     The  position  of  a  point  P  in  a  line  is  deter- 
mined by  its  distance  OP=x  from  some  fixed  point  or  origin, 
O,  assumed  in  the  line,  the  length  x  being  taken  with  the 
proper  sign  to  express  the  sense  (say  forward  or  backward,  to 
the  right  or  to  the  left)  in  which  it  is  to  be  measured  on  the 
line.     This  sense  is  also  indicated  by  the  order  of  the  letters,  so 
that  PO=-OP,  and  OP+PO  =  o. 

The  position  of  a  point  in  a  line  is  thus  fully  determined  by 
a  single  algebraical  quantity  or  co-ordinate ;  viz.  by  its  abscissa 
x=OP. 

7.  Let  the  point  P  move  in  the  line  from  any  initial  position 
PQ  (Fig.  i)  to  any  other  position  Pv  and  let  OP0=xQ,  OP^xv 


This  change  of  position,  or  displacement,  is  fully  determined 
by  the  distance  PQPl=^1—  x^  traversed  by  the  point. 

Now  let  this  displacement  PQPl  be  followed  by  another  dis- 
placement in  the  same  line,  from  Pl  to  P2,  in  the  same  sense  as 


4  GEOMETRY   OF  MOTION.  [8. 

the  former,  or  in  the  opposite  sense.  In  either  case  the  total, 
or  resultant,  displacement  is  the  algebraic  sum  of  the  two  dis- 
placements PoPv  P\P%>  which  are  called  its  components;  i.e. 
we  have  />0/>2  = /y^  + /y>2>  or  /y\  + /y>2  + /y>0  ==  o,  what- 
ever may  be  the  positions  of  the  points  PQ,  PI}  P%  in  the  line. 

This  reasoning  is  easily  extended  to  any  number  of  compo- 
nent displacements ;  that  is,  the  resultant  of  any  number  of 
consecutive  displacements  of  a  point  in  a  line  is  a  single  displace- 
ment equal  to  the  algebraic  sum  of  the  components. 

Similar  considerations  apply  to  the  motion  of  a  point  in  a 
curved  line  provided  the  displacements  be  always  measured 
along  the  curve. 

8.  Let  us  next  consider  the  motion  of  a  rigid  body.     The 
term  rigid  body,  or  simply  body,  is  used  in  kinematics  to  denote 
a  figure  of  invariable  size  or  shape,  or  an  aggregate  of  points 
whose  distances  from  each  other  remain  unchanged.     Examples 
are  :  a  segment  of  a  straight  line,  a  triangle,  a  cube,  an  ellipsoid, 
etc. 

Imagine  such  a  body  M  brought  in  any  manner  from  some 
initial  position  MQ  into  any  other  position  Mv  This  displace- 
ment M0Ml  is  determined  by  the  displacements  of  the  various 
points  of  the  body.  We  shall  see  that,  even  in  the  most  general 
case,  the  displacements  of  three  points  of  the  body  determine 
those  of  all  other  points,  and  consequently  the  displacement  of 
the  whole  body. 

There  are,  however,  two  special  cases  of  motion,  translation 
and  rotation,  in  which  the  displacement  of  the  body  is  fully 
determined  by  the  displacement  of  a  single  point :  such  motions 
can  be  called  linear.  There  is  also  a  class  of  motions  deter- 
mined by  the  displacements  of  only  two  points  of  the  body : 
this  is  called  plane  motion. 

9.  The  displacement  of  a  rigid  body  is  called  a  translation 
when  the  displacements  of  all  of  its  points  are  parallel  and  equal. 
It  is  evident  that  in  this  case  the  displacement  of  any  one 


ii.]  LINEAR   MOTION.  5 

point  of  the  body  fully  represents  the  displacement  of  the 
whole  body.  The  translation  MQM1  of  a  rigid  body  M  from  a 
position  MQ  to  a  position  M^  is  therefore  measured  by  the 
rectilinear  segment  PQP1  that  represents  the  displacement  of 
any  point  P  of  the  body  M. 

Two  or  more  consecutive  translations  of  a  rigid  body  in  the 
same  direction  produce  a  resultant  displacement  of  translation 
equal  to  the  algebraic  sum  of  the  components. 

10.  When  a  rigid  body  has  two  of  its  points  fixed,  the  only 
motion  it  can  have  is  a  rotation  about  the  line  joining  the  fixed 
points  as  axis.     In  a  motion  of  rotation  all  points  of  the  body 
excepting  those  on  the  axis  describe  arcs  of  circles  whose  centres 
lie  on  the  axis  while  the  points  on  the  axis  are  at  rest. 

The  different  positions  of  a  rotating  body  may  be  referred 
to  any  fixed  plane  passing  through  the  axis  of  rotation.  Any 
plane  of  the  body  passing  likewise  through  the  axis  will  make 
with  the  fixed  plane  an  angle  6  which  varies  in  the  course 
of  the  motion.  This  angle,  taken  with  the  proper  sign,  fully 
determines  the  positions  of  the  body. 

Let  the  body  rotate  from  a  position  00  to  a  position  6^ ;  the 
angle  0-^  —  0^  measures  the  corresponding  displacement,  or  the 
rotation,  just  as  (Art.  7)  the  distance  PQPl=^l—  x§  measures 
the  displacement  of  a  point,  and  hence  (Art.  9)  the  translation 
of  a  rigid  body. 

Two  or  more  consecutive  rotations  of  a  rigid  body  about  the 
same  axis  give  a  resultant  rotation  whose  angle  is  the  algebraic 
sum  of  the  angles  of  the  component  rotations. 

11.  The  particular  case  when  the  rigid  body  is  a  plane  figure 
whose  motion  is  confined  to  its  plane  deserves  special  mention. 
If  one  point  of  such  a  figure  be  fixed,  the  figure  can  only  have 
a  motion  of  rotation,  every  other  point  of  the  figure  describing 
an  arc  of  a  circle  whose  centre  is  the  fixed  point.     This  point 
is  therefore  called  the  centre  of  rotation.     The  positions  of  the 
figure  are  given  by  the  angle  that  any  line  of  the  figure  passing 


6  GEOMETRY   OF   MOTION.  [12. 

through  the  centre  makes  with  any  fixed  line  through  the  centre 
in  the  plane. 

12.  We  have  seen  that  a  translation  as  well  as  a  rotation  is 
measured  by  a  single  algebraical  quantity,  the  translation  by  a 
distance,  the  rotation  by  an  angle.  This  is  the  reason  why 
such  motions  may  be  called  linear  or  of  one  dimension.  The 
two  fundamental  forms  of  motion,  translation  and  rotation,  are 
thus  seen  to  correspond  to  the  two  fundamental  magnitudes  of 
metrical  geometry,  viz.  distance  and  angle. 

It  is  to  be  noticed  that  both  for  translations  in  the  same 
direction  and  for  rotations  about  the  same  axis  the  resultant 
displacement  is  found  by  algebraic  addition  of  the  components, 
not  only  when  the  components  are  consecutive  motions,  but  even 
when  they  are  simultaneous.  Thus  we  may  imagine  a  point  P 
displaced  by  the  amount  P^P^  along  a  straight  line  while  this 
line  itself  is  moved  along  in  its  own  direction  by  an  amount 
Q^Qv  The  resultant  displacement  of  P  is  the  algebraic  sum 


13.  Translations   being   measured   by  distances  or  lengths, 
and  rotations  by  angles,  we  need  in  mechanics  a  unit  of  length 
and  a  unit  of  angle. 

The  two  most  important  systems  of  measurement  are  the 
C.  G.  S.  (i.e.  centimetre-gramme-second)  system,  and  the  F.  P.  S. 
(i.e.  foot-pound-second)  system.  The  former  is  frequently 
called  the  scientific  system  ;  it  is  based  on  the  international 
or  metric  system  of  weights  and  measures.  The  F.  P.  S.,  or 
British  system,  is  still  used  in  England  and  the  United  States 
almost  universally  in  engineering  practice.* 

14.  The  unit  of  length  in  the  C.  G.  S.  system  is  the  centimetre 
(cm.),  -i.e.  YOU  of  the  metre.     The  original  standard  metre  is  a 


*  For  fuller  information  on  all  questions  relating  to  standards  and  units  see 
J.  D.  EVERETT,  Illustrations  of  the  C-  G-.  S.  svsten:  of  units  with  tables  of  physical 
constants;  London,  Macmillan,  1891. 


IS-]  LINEAR   MOTION.  7 

platinum  bar  preserved  in  the  Palais  des  Archives  in  Paris,  a 
legalized  copy  of  which  has  been  deposited  at  Washington, 
D.C.  The  metre  can  be  defined  as  the  distance  between  two 
marks  on  the  standard  metre  when  at  a  temperature  of  o°  C. 

In  the  F.  P.  S.  system,  the  unit  of  length  is  the  foot,  i.e.  -J-  of 
the  standard  yard.  The  original  British  standard  yard  is  a 
bronze  bar  preserved  in  London.  For  the  United  States  the 
yard  is  defined  as  the  distance  between  the  twenty-seventh  and 
sixty-third  divisions  of  the  brass  standard  yard  kept  in  the 
Bureau  of  Weights  and  Measures  at  Washington,  when  the  bar 
is  at  a  temperature  of  i6|°  C.  or  62°  F. 

The  relation  between  these  two  fundamental  units  of  length 
is,  according  to  the  United  States  Coast  and  Geodetic  Survey 
Bulletin  No.  9,  1889, 

I  cm.  =  0.032  808  2  ft. 
For  practical  use  we  have  the  following  approximate  relations  : 

i  m.  =  3.2809  ft.,  i  ft.  =  30.48  cm., 

i  cm.  =  0.3937  in.,  i  in.  =  2.54  cm. 

15.  The  unit  of  angle  is  either  the  degree,  i.e.  -^  of  one 
revolution,  or  the  radian,  i.e.  the  angle  measured  by  an  arc 
whose  length  is  equal  to  the  radius. 

If  «  be  any  angle  expressed  in  radians,  and  «°,  a',  a"  the  same 
angle  expressed  respectively  in  degrees,  minutes,  seconds,  we 
have  the  relations 

7T  o  T  i  7T  it 

a= •  cc°  = •  a'  =  — — •  or, 

I 80  I08OO  648000 

or  a  =  o.Oi/  45 301°  =  o.OOO  29 1  <*'  =  0.000004  8 5  a". 


GEOMETRY   OF   MOTION. 


[16. 


II.     Plane  Motion. 

16.  The  position  of  a  plane  figure  in  its  plane  is  fully  deter- 
mined  by  the  positions  of   any  two  of   its   points  since  every 
other  point  of  the  figure  forms  with  these  two  points  an  invari- 
able triangle.     But  the  position  of  the  figure  can  of  course  be 
determined  in  other  ways  ;  for  instance,  by  the  position  of  one 
point  and  that  of  a  line  of  the  figure  passing  through  the  point  ;. 
or  by  the  position  of  two  lines  of  the  figure. 

17.  Let  us  now  consider  the  motion  of  a  plane  figure  F  in  its. 
plane  from   any  initial  position  FQ   to   any  other    position  F^ 
The  displacement  FQF1  can  be  brought  about  in  various  ways. 

Thus,  it  would  be  suffi- 
cient to  bring  any  two 
points  A,  B  (Fig.  2)  of 
the  figure  F  from  their 
initial  positions  AQ,  BQ  in 
FQ  to  their  final  positions 
'Bi  Av  Bl  in  Fv  This  can 
be  done,  for  instance,  by 
first  giving  the  whole  fig- 
ure a  translation  through 
a  distance  AQA1  and  then 
a  rotation  by  an  angle 
^  ;  or  by  such  a  rota- 


Fig. 2. 


and 


equal  to  the  angle  between  AQ 
tion  followed  by  the  translation. 

Instead  of  A  we  might  have  selected  any  other  point  of  the 
figure.  But  it  is  important  to  notice  that  the  angle  of  rotation 
required  for  a  given  displacement  FQFl  is  always  the  same,  while 
the  translation  will  differ  according  to  the  point  selected  as 
centre. 

18.  This  leads  us  to  inquire  whether  the  centre  of  rotation 
cannot  be  so  selected  as  to  reduce  the  translation  to  zero. 
Now  any  rotation  that  is  to  bring  A  from  AQ  to  A1  must  have 


20.] 


PLANE   MOTION. 


its  centre  on  the  perpendicular  bisector  of  A^A^\  similarly  for 
B.  Hence  the  intersection  C  of  the  perpendicular  bisectors  of 
AQA1  and  B^B^  is  the  only  point  by  rotation  about  which  both 
A  and  B  can  be  brought  from  their  initial  to  their  final  posi- 
tions. That  they  actually  are  so  brought  follows  at  once  from 
the  equality  of  the  angles  A^CBQ  and  A1C^1  (and  hence  of  the 
angles  A^CAl  and  B^CB^)  which  are  homologous  angles  in  the 
equal  triangles  A^CBQ  and  A^CBV 

We  thus  have  the  proposition  :  Any  displacement  of  an  inva- 
riable plane  figure  in  its  plane  can  be  brought  about  by  a  single 
rotation  about  a  certain  point  which  we  may  call  the  centre  of 
the  displacement. 

19.   The  construction  of  the  centre  C  given  in  the  preceding 
article  becomes  impossible  when  the  bisectors  coincide  (Fig.,  3) 
and  when  they  are  parallel  (Fig.  4). 
In  the  former   case,   C  is  readily 
found  as  the  intersection  of  A^B§ 
and  A^BV    In  the  latter,  i.e.  when- 
ever A^A^B^B^  the  centre  lies  at 
infinity,  and  the  rotation  becomes 
a  translation. 


Any  translation  may  therefore  be  regarded  as  a  rotation  about 
a  centre  at  infinity. 

20.  Let  the  figure  F  pass  through  a  series  of  displacements 
FQFlt  F-)Fy  ...  Fn_±Fn.  Each  displacement  has  its  angle  and 
its  centre.  If  the  successive  positions  FQ,  Fv  ...  Fn  of  the  figure 


IO 


GEOMETRY   OF   MOTION. 


[21. 


are  taken  each  very  near  the  preceding  one,  the  angles  of  rota- 
tion will  be  very  small,  and  the  successive  centres  Cv  C2,  ... 
Cn  will  follow  each  other  very  closely.  In  the  limit,  i.e.  when  for 
the  series  of  finite  displacements  we  substitute  a  continuous 
motion  of  the  figure,  the  centres  C  will  form  a  continuous  curve 
(c)  and  the  angles  become  the  infinitely  small  angles  between 
the  successive  normals  to  the  paths  described  by  the  points  of 
the  figure.  The  point  C  about  which  the  figure  rotates  in  any 
one  of  its  positions  during  the  motion  is  now  called  the  instan- 
taneous centre;  the  locus  of  the  centres,  that  is  the  curve  (c),  is 
called  the  centrode,  or  path  of  the  centre.  It  is  apparent  that 
in  any  position  of  the  moving  figure  the  normals  to  the  paths  of 
all  its  points  must  pass  through  the  instantaneous  centre,  and  the 
direction  of  motion  of  any  such  point  is  therefore  at  right  angles 
to  the  line  joining  it  to  the  centre. 

21.   The  centres  C  are  points  of  the  fixed  plane  in  which  the 
motion  of  the  figure  F  takes  place.     But  in  any  position  F1  of 

this  figure  some  point 
C\  of  F  will  coincide 
with  the  point  C±  of  the 
fixed  plane.  Thus,  in 
the  case  of  finite  dis- 
placements (Fig.  5),  let 
the  figure  F  begin  its 
motion  with  a  rotation 
of  angle  Ol  about  a  point 
C-L  of  the  fixed  plane ; 
let  C\  be  the  point  of 
the  moving  figure  that 
coincides  during  this 


Fig.  5. 


rotation  with  Cv 


The  next  rotation,  of  angle  02,  takes  place  about  a  point  C2 
of  the  fixed  plane.  The  point  of  the  moving  figure  that  now 
coincides  with  C2  was  brought  into  the  position  C2  by  the  pre- 


23-]  PLANE   MOTION.  U 

ceding  rotation.  Its  original  position  is  therefore  obtained  by 
turning  6\(72  back  by  an  angle  —6l  into  the  position  C^C <%. 
The  rotation  of  angle  02  about  (72  brings  a  new  point  (73  of 
the  moving  figure  to  coincidence  with  the  fixed  centre  C3 ;  and 
the  original  position  C'3  of  this  point  can  be  determined  by 
first  turning  C2CB  back  about  C2  by  an  angle  —  02  into  the 
position  C2D,  and  then  turning  the  broken  line  C^C^D  by  a 
rotation  of  angle  —  0:  about  6\  back  into  the  position  C\C\C\. 
Continuing  this  process  we  obtain,  besides  the  broken  line 
C1C2CB...  formed  by  joining  the  successive  centres  of  rotation 
in  the  fixed  plane,  a  broken  line  C\CyC\...  in  the  moving 
figure  formed  by  joining  those  points  of  this  figure  which  in  the 
course  of  the  motion  come  to  coincide  with  the  fixed  centres. 
The  whole  motion  may  be  regarded  as  a  kind  of  rolling  of  the 
broken  line  C\C\C\ . . .  over  the  broken  line  €<£<£%  — 

22.  In  the  case  of  continuous  motion  each  of  the  broken  lines 
becomes  a  curve,  and  we  have  actual  rolling  of  the  curve  (cf),  or 
body  centrode,   over  the  curve  (c),  or  space  centroder    The  con- 
tinuous motion  of  an  invariable  plane  figure  in  its  plane  may 
therefore  always  be  produced  by  the  rolling  (without  sliding)  of 
the  body  centrode  over  the  space  centrode.     The  point  of  contact 
of  the  two  curves  is  of  course  the  instantaneous  centre. 

23.  It  appears  from  the  preceding  articles  that  the  continuous 
motion  of  a  plane  figure  in  its  plane  is  fully  determined  if  we 
know  the  centre  of  rotation  for  every  position  of  the  figure. 
This  centre  can  be  found  as  the  intersection  of  the  normals  of 
the  paths  of  any  two  points  of  the  figure,  so  that  the  motion 
of  the  figure  will  be  known  if  the  paths  of  any  two  of  its  points 
are  given.     This,  however,  is  only  one  out  of  many  ways  of 
determining  plane  motion  by  two  conditions. 

Thus  the  motion  may  be  determined  by  the  condition  that  a 
curve  of  the  moving  figure  should  remain  in  contact  with  two 
fixed  curves.  In  this  case  the  instantaneous  centre  is  found  as 
the  intersection  of  the  common  normals  at  the  points  of  contact. 


12 


GEOMETRY   OF  MOTION. 


[24- 


The  condition  that  a  curve  of  the  moving  figure  should  always 
pass  through  a  fixed  point  may  be  regarded  as  a  special  case  of 
the  condition  just  mentioned,  one  of  the  fixed  curves  being 
reduced  to  a  point. 

24.  Any  curve  of  the  moving  figure  forms  during  the  motion 
an  envelope,  the  points  of  the  envelope  being  the  intersections 
of  the  .successive  infinitely  near  positions  of  the  moving  curve. 
Let  /,  /'  be  two  such  successive  positions  of  the  curve,  A  their 
intersection,   C  the  instantaneous   centre ;  then   CA  is  perpen- 
dicular to  /  as  well  as  to  /',  and  hence  to  the  envelope.     The 
envelope  can  therefore  be  constructed  by  letting  fall  normals 
from  the  instantaneous  centres  on  the  corresponding  positions 
of  the  generating  curve. 

25.  The  following   examples  will   illustrate   the   method   of 
finding  the  centrodes  and  the  path  of  any  point  of  the  moving 
figure  in  plane  motion. 

Elliptic  motion  :  Two  points  of  a  plane  figure  move  along  two 
fixed  lines  that  are  at  right  angles  to  each  other. 

Let  Ay  B  (Fig.  6)  be  the  points  moving  on  the  lines  Ox,  Oy ; 
the  perpendiculars  to  these  lines  erected  at  A  and  B  intersect 
at  the  instantaneous  centre  C.  Denoting  by  2  a  the  invariable 

distance  of  A  and  B,  we  have 
OC—AB=2a  for  all  posi- 
tions of  the  moving  figure. 
The  fixed  centrode  (c)  is 
therefore  a  circle  of  radius 
2  a  described  about  the  in- 
tersection O  of  the  fixed 


lines. 


Fig.  6. 


To  find  the  body  centrode 
(c1)  we  must  construct  the 
triangle  ABC  for  all  possible 
positions  of  AB.  As  BCA  is  always  a  right  angle,  the  body 
centrode  will  be  a  circle  described  on  AB  as  diameter.  Hence 


26.] 


PLANE    MOTION. 


the  whole  motion  can  be  produced  by  the  rolling  of  a  circle  of 
radius  a  within  a  circle  of  radius  2  a. 

The  student  is  advised  to  carefully  carry  out  the  construc- 
tions indicated  in  this  as  well  as  the  following  problems.  Thus, 
in  the  present  case,  draw  the  moving  figure,  i.e.  the  line  AB, 
in  a  number  of  its  successive  positions  in  each  of  the  four 
quadrants,  and  construct  the  instantaneous  centre  C  in  every 
case.  This  gives  a  number  of  points  of  the  space  centrode. 
Then  take  any  one  position  of  AB  and  transfer  to  it  as  base 
all  the  triangles  ABC  previously  constructed.  The  vertices 
of  these  triangles  all  lie  on  the  body  centrode. 

26.  To  find  the  equation  of  the  path  of  any  point  P  of  the 
moving  figure,  let  this 
point  be  referred  to  a  co- 
ordinate  system  fixed  in, 
and  moving  with,  the  fig- 
ure (Fig.  7)  ;  let  the  mid- 
dle point  Or  of  AB  be 
the  origin,  and  O'A  the 
.axis  O'x',  of  this  system. 
Then  the  co-ordinates  x', 


Fi 


j/f  of  P  in  this  moving 

system  are  connected  with 

its  co-ordinates  x,  y  in  the  fixed  system  Ox,  Oy  by  the  following 

equations, 


'  cos<£, 


y—(a—x!] 


where  <£  is  the  angle  OAB  that  determines  the  instantaneous 
position  of  AB.  Solving  these  equations  for  sin<£  and  cos0, 
squaring  and  adding,  we  find  for  the  equation  of  the  path  of  P 


_ 

- 


Or         ^~ 


!4  GEOMETRY   OF   MOTION, 

which  represents  an  ellipse,  since  the  determinant 

a—  xf)z+yfz,         —  2ayf 
-2ay'y 


[27. 


is  necessarily  positive. 

In  general,  therefore,  the  points  of  the  figure  describe 
ellipses  ;  O'  describes  a  circle  ;  A  and  B  describe  straight  lines, 
and  so  does  every  point  on  the  circle  of  diameter  AB.  It  is 
this  fact  that  by  rolling  a  circle  within  a  circle  of  double  diam- 
eter the  points  of  the  smaller  circle  are  made  to  describe  seg- 
ments of  straight  lines,  which  makes  this  form  of  motion  of 
practical  importance  :  it  may  serve  to  transform  circular  into 
rectilinear  motion. 

27.  Elliptic  Motion  (continued)  :  Two  points  A,  B  of  a  plane 
figure  move  along  two  fixed  lines  inclined  to  each  other  at  an 
angle  a>  (Fig.  8). 


Fig.  8. 


This  case  is  readily  reduced  to  the  preceding  one.  The 
instantaneous  centre  is  found  as  before ;  its  distance  OC  from 
the  intersection  of  the  fixed  lines  OA,  OB  is  again  constant 


28.] 


PLANE  MOTION. 


and  =AB/sm  a  ;  for  O,  A,  C,  B  all  lie  on  a  circle  whose  centre 
O'  bisects  OC;  hence 

The  motion  is  therefore  produced  by  the  rolling  of  this  circle 
of  diameter  AB/sm  o>  within  a  circle  of  twice  this  diameter 
described  about  O ;  it  is  not  essentially  different  from  the  pre- 
ceding case  (Art.  26).  This  will  also  be  seen  if  we  take  OA  as 
axis  of  x,  the  perpendicular  to  it  through  O  as  axis  of  y.  This 
perpendicular  Oy  intersects  the  circle  OAB  in  a  point  B',  which 
is  the  end  of  the  diameter  AO'B'  and  moves  along  Oy  during 
the  motion.  The  points  A,  B1  of  the  figure  move,  therefore, 
along  the  rectangular  lines  Ox,  Oy,  just  as  in  the  problem  of 
Art.  26. 

28.  Connecting  Rod  Motion :  One  point  A  of  the  figure  describes 
a  circle,  while  another  point  B  moves  on  a  straight  line,  passing 
through  the  centre  O  of  the  circle  (Fig.  9). 


Fig.  9. 


With  OB  as  polar  axis,  the  equation  of  the  fixed  centrode  is 
r*  cos2  d  -  2  ar  cos2  6  +  a2  =  /2. 

This,  as  well  as  the  equation  of  the  body  centrode,  is  of  the 
sixth  degree  in  rectangular  Cartesian  co-ordinates.  But  the 
graphical  construction  presents  no  difficulties. 


i6 


GEOMETRY   OF   MOTION. 


[29. 


29.    Conchoidal  Motion :  A  point  A  of  the  figure  moves  along  a 
fixed  straight  line  1,  while  a  line  of  the  figure,  1',  containing  the 

point  A  always  passes 
through  a  fixed  point  B 
(Fig.  10). 

The  fixed  point  B  may 
be  regarded  as  a  circle 
of  infinitely  small  radius, 
which  the  line  /'  is  to 
touch.  The  instantane- 
ous centre  is  therefore 
the  intersection  C  of  the 
perpendiculars  erected  at  A  on  /  and  at  B  on  I'. 

The  fixed  centrode  is  a  parabola  whose  vertex  is  B.  To 
prove  this  we  take  the  fixed  line  /  as  axis  of  y,  the  perpendicular 
OB  to  it  drawn  through  the  fixed  point  B  as  axis  of  x.  Then, 
putting  ^.OBA=(j>  and  OB  =  a,  we  have  for  the  co-ordinates 
ofC 


10. 


y—a  tan<£ ; 

hence  x—a=y2/a,  or,  for  B  as  origin  and  parallel  axes,  y*=ax. 

The  equation  of  the  body  centrode,  for  OB,  OA  as  axes  of  x 
and;j/,  is  a\x2+y*)=x*,  or  r  cos2  Q— a. 

The  points  of  /'  can  easily  be  shown  to  describe  conchoids, 
whence  the  name  of  this  form  of  plane  motion. 

30.  The  results  obtained  in  the  preceding  articles  for  the 
motion  of  a  plane  figure  in  its  plane  apply  directly  to  the  motion 
of  a  rigid  body,  if  any  one  point  of  the  body  describes  a  plane 
curve  while  a  line  of  the  body  remains  parallel  to  itself.  For  in 
this  case  all  points  of  the  body  move  in  parallel  planes,  and  the 
motion  in  any  one  of  these  planes  determines  the  motion  of  the 
whole  figure. 

The  only  modifications  required  would  be  that  instead  of  an 
instantaneous  centre  we  should  have  an  instantaneous  axis,  viz. : 


3i.]  PLANE   MOTION.  I7 

the  perpendicular  to  the  plane  of  motion  of  any  point  through 
the  centre  of  motion  of  this  point ;  and  that  the  centrodes  are 
now  not  curves,  but  cylindrical  surfaces  rolling  one  upon  the 
other. 

31.    Exercises. 

(1)  Show  how  to  find  the  direction  of  motion  of  any  point  /'rigidly 
connected  with  the  connecting  rod  of  a  steam  engine. 

(2)  A  wheel  rolls  on  a  straight  track;   find  the  direction  of  motion 
of  any  point  on  its  rim.     What  are  the  centrodes  in  this  case  ? 

(3)  Show  how  to  construct  the  normal  at  any  point  of  a  conchoid. 

(4)  Find  the  equation  of  the  fixed   centrode  when  a  line  V  of  a 
plane  figure  always  touches  a  fixed  circle  O,  while  a  point  A  of  V  moves 
along  a  fixed  line /. 

(5)  Show  that,  in  (4),  the  fixed  centrode  is  a  parabola  when  the 
fixed  circle  touches  the  fixed  line. 

(6)  Two  straight  lines  /f,  /"  of  a  plane  figure  constantly  pass  each 
through  a  fixed  point  O',  O" ;  investigate  the  motion. 

(7)  Four  straight  rods  are  jointed  so  as  to  form  a  plane  quadrilateral 
ABDE  with  invariable  sides  and  variable  angles.     One  side  AB  being 
fixed,  investigate  the  motion  of  the  opposite  side ;  construct  the  cen- 
trodes graphically. 

(8)  Let  a  straight  line  /  in  a  fixed   plane  be  brought   by  a  finite 
displacement  from  an  initial  position  /0  into  a  final  position  4 ;  and  let 
P  be  any  point  of/,  PQ  its  initial  position  (in  /0),  Pl  its  final  position 
(in  4) .     Then  the  following  propositions  can  be  proved  : 

(a)  The  middle  points  of  the  displacements  P^  of  all  points  P  of 
/  lie  in  a  straight  line  ; 

(£)    the  lines  /o/i  envelop  a  parabola ; 

(<:)  the  projections  of  the  displacements  P^Pi  on  the  line  joining 
their  middle  points  are  all  equal ; 

(d)  if  /  have  a  continuous  motion  in  the  plane,  the  tangents  to  the 
paths  of  all  its  points  envelop  a  parabola  of  which  the  instantaneous 
centre  is  the  focus  and  /  the  tangent  at  the  vertex. 

PART   I — 2 


1 8  GEOMETRY   OF  MOTION.  [32. 

III.    Spherical  Motion. 

32.  The  motion  of  a  spherical  figure  of  invariable  form  on  its 
sphere  presents  a  close  analogy  to  plane  motion  ;  in  fact,  plane 
motion   is   but   a   special    case   of    spherical    motion,    since    a 
plane  may  be  regarded  as  a  sphere  of  infinite  radius. 

33.  By  a  generalization  similar  to  that  of  Art.  30,  the  study 
of  the  motion  of  a  spherical  figure  on  its  sphere  leads  directly  to 
the  laws  of  motion  of  a  rigid  body  having  one  fixed  pgint.     For 
the  motion  of  such  a  body  is  evidently  determined  by  the  spheri- 
cal motion  on  any  sphere  described  about  the  fixed  point. 

34.  Let  us  consider  any  two  positions  FQ  and  F1  of  a  spheri- 
cal figure  Fon  its  sphere,  and  let  O  be  the  centre  of  the  sphere. 
Just  as  in  the  case  of  plane  motion  (Art.  18)  the  displacement 
FQF1  can  always  be  brought  about  by  a  single  rotation  about  a 
point  C  on  the  sphere,  or  what  amounts  to  the  same,  by  a  single 
rotation  about  the  axis  OC.     The  proof   is  strictly  analogous 

to  that  given  in  Art.  18.  We 
first  remark  that  the  position  of 
the  figure  on  the  sphere  is  fully 
determined  by  the  position  of 
two  of  its  points,  say  A  and  B 
(Fig.  n),  since  any  third  point 
forms  with  these  an  invariable 
spherical  triangle.  Let  A0,  B^ 
be  the  positions  of  A,  B  in  FQ ; 
Av  B^  their  positions  in  Fl ; 
and  draw  the  great  circles  A0Al 
and  B^BV  Their  perpendicular 
bisectors  intersect  in  two  points  C,  D  which  are  the  ends  of  a 
diameter  of  the  sphere.  CD  is  the  axis  of  the  displacement 
FQFl}  and  the  angle  A^CA^  or  B^CB^  gives  the  angle  of  the 
displacement. 


37-]  SCREW   MOTION.  !9 

35.  If  we  consider  a  series  of  positions  of  the  moving  figure, 
F0,  Fv  Fy  . . .,  we  obtain  a  series  of  axes  of  rotation,  say  cv  cz,  . . .  ; 
and  in  the  limit  when  these  positions  follow  one  another  at 
infinitely  near  intervals,  the  axes  cv  cy  ...  will  form  a  cone  fixed 
in  space,  with  the  vertex  at  the  centre  O  of  the  sphere.     The 
points  Cv  C2,  ...  where  these  axes  intersect  the  sphere  form  a 
curve  (c)  on  the  fixed  sphere,  while  the  points  C'v  C'%,  ...  of  the 
moving  figure  with  which  these  fixed  points  come  to  coincide 
form  a  spherical  curve  (c')  invariably  connected  with  the  moving 
figure.     The  whole  motion  may  be  produced  by  the  rolling  of 
the  curve  (c'}  over  the  curve  (c),  or  also  by  the  rolling  of  the 
corresponding  cones  one  over  the  other.      We  have  thus  the 
proposition  that  any  continuous  motion  of  a  rigid  body  having  a 
fixed  point  can  be  produced  by  the  rolling  of  a  cone  fixed  in  the 
body  on  a  fixed  cone,  the  vertices  of  both  cones  being  at  the  fixed 
point. 

IV.     Screw  Motion. 

36.  The  position  of  a  rigid  body  in  space  is  fully  determined 
by  the  position  of  any  three  of  its  points  not  situated  in  the 
same  straight  line.     For  any  fourth  point  of  the  body  will  form 
an  invariable  tetrahedron  with   these   three  points.      As  two 
points  determine  a  straight  line,  the  position  of   a  rigid  body 
may  also  be  given  by  the  position  of  a  point  and  line  or  by 
the  positions  of  two  intersecting  or  parallel  lines  of  the  body. 

37.  The   position  of  a  point  being  determined  by  its  three 
co-ordinates  requires  three  conditions  to  be  fixed.     A  point  is 
therefore  said  to  have  three  degrees  of  freedom  when  its  position 
is    not    subject   to   any  conditions.     One  conditional  equation 
between  its  co-ordinates  restricts  the  point  to  the  surface  repre- 
sented by  that  equation  ;  the  point  is  then  said  to  have  but 
two  degrees  of   freedom  and  one  constraint.     Two  conditions 
would  restrict  the  point  to  a  line,  the  curve  of  intersection  of 
the   two  surfaces  represented  by  the  equations  of  condition ; 


20  GEOMETRY   OF   MOTION.  [38. 

the  point  has  then  but  one  degree  of  freedom  and  two  con- 
straints. 

A  rigid  body  that  is  perfectly  free  to  move  has  six  degrees  of 
freedom.  For  we  have  seen  that  its  position  is  fully  determined 
when  three  of  its  points  not  in  the  same  line  are  fixed.  The 
nine  co-ordinates  of  these  points  are,  however,  not  indepen- 
dent ;  they  are  connected  by  the  three  equations  expressing 
that  the  three  distances  between  the  three  points  are  invariable. 
Thus  the  number  of  independent  conditions  is  9  —  3  =  6. 

A  rigid  body  with  one  fixed  point  has  three  degrees  of  freedom 
and  therefore  three  constraints.  For  it  takes  two  more  points, 
i.e.  six  co-ordinates,  to  fix  the  position  of  the  body ;  and  the 
distances  of  these  two  points  from  each  other  and  from  the 
fixed  point  being  invariable,  there  are  again  three  conditional 
equations  to  which  the  six  co-ordinates  are  subject.  The  three 
co-ordinates  of  the  fixed  point  may  be  regarded  as  the  three 
constraints. 

A  rigid  body  with  two  fixed  points,  i.e.,  with  a  fixed  axis,  has 
one  degree  of  freedom,  and  five  constraints.  Indeed,  the  six 
co-ordinates  of  the  two  fixed  points  are  equivalent  to  five  con- 
straining conditions,  since  the  distance  of  these  two  points  is 
invariable.* 

38.  Let  us  now  consider  any  two  positions  MQ,  M±  of  a  rigid 
body  M,  given  by  the  positions  AQ,  BQ,  CQ  and  Av  Bv  £\  of 
three  points  A,  B,  C  of  the  body.  The  displacement  MQMl 
can  be  effected  in  various  ways.  Thus  we  might  for  instance 
begin  by  giving  the  whole  body  a  translation  equal  to  AQAl 
which  would  bring  the  point  A  to  its  final  position  while  all 
other  points  of  the  body  would  be  displaced  by  distances  par- 
allel and  equal  to  AQAr  As  the  body  has  now  one  of  its 
points,  A,  in  its  final  position,  it  will  (by  Art.  34)  require  only 


*  Interesting  remarks  on  the  mechanical  means  of  producing  constraints  of 
various  degrees  will  be  found  in  THOMSON  and  TAIT,  Natural  philosophy,  London, 
Macmillan,  new  edition,  1879,  Art.  195  sq.  (Part  I.,  p.  149). 


40.]  SCREW   MOTION.  21 

a  single  rotation  about  a  certain  axis  passing  through  this  point 
to  bring  the  whole  body  into  its  final  position.  It  thus  appears 
that  any  displacement  of  a  rigid  body  can  be  effected  by  sub- 
jecting the  body  first  to  a  translation  and  then  to  a  rotation 
(or  vice  versa,  as  is  easily  seen) ;  and  this  can  be  done  in  an 
infinite  number  of  ways,  as  the  displacement  of  any  point  of 
the  body  may  be  selected  for  the  translation. 

39.  It  is  to  be  noticed  that  for  all  these  different  ways  of 
effecting  the  displacement  M^Ml  the  direction  of  the  axis  of 
rotation  and  the  angle  of  rotation  are  the  same.     To  see  this 
more  clearly,  let  the  displacement  be  effected  first  by  the  trans- 
lation AQA1  and  a  rotation  of  angle  a  about  the  axis  a±  passing 
through  Al ;  and  then  let  the  same  displacement  be  produced 
by  the  translation  B^B^  of  some  other  point  B  and  a  rotation  qf 
angle  /3  about  an  axis  b±  passing  through  B^.    We  wish  to  show 
that  a^  and  b±  are  parallel  and  that  the  angles  a  and  ft  are  equal. 

Consider  a  plane  TT  of  the  rigid  body  which  in  its  original 
position  TTO  is  perpendicular  to  the  axis  av  The  translation 
AQAl  transfers  it  into  a  parallel  position  and  the  rotation  a  about 
a1  turns  it  in  itself  into  its  final  position  TTJ  ;  hence  TTO  and  TTJ 
are  parallel.  The  translation  B0B1  likewise  moves  TT  into  a 
position  parallel  to  the  original  one  ;  and  as  its  final  position, 
irv  is  parallel  to  TTO,  the  axis  of  rotation  b±  must  necessarily  be 
perpendicular  to  TTO  and  7r1}  that  is  bl  must  be  parallel  to  a^. 

Again,  any  straight  line  /  in  TT  remains  parallel  to  its  original 
position  /0  after  the  translations  A0A1  and  B^BV  Its  change  of 
direction  is  due  to  the  rotations  alone  ;  the  angle  of  rotation 
must  therefore  be  the  same  for  both  rotations,  viz.  equal  to  the 
angle  (/Q/J)  formed  by  the  initial  and  final  positions  of  the  line  /. 

40.  Among  the  different  combinations  of  a  translation  with 
a  rotation  effecting   the  displacement   MQMl   there    is    one  of 
particular  importance ;  it  is  that  for  which  the  axis  of  rotation 
is  parallel  to  the  translation. 

Let  us  again  consider  the  plane  TT  perpendicular  to  the  com- 


22  GEOMETRY    OF   MOTION.  [41. 

mon  direction  of  the  axes  of  rotation.  To  bring  any  three 
points  of  this  plane  into  their  final  position  it  is  only  necessary 
to  give  the  body  a  translation  at  right  angles  to  TT  such  as  to 
bring  TT  into  its  final  position  and  then  to  add  the  necessary 
rotation  for  plane  motion. 

We  have  therefore  the  important  proposition  that  it  is  always 
possible  to  bring  a  rigid  body  M  from  any  position  M0  into  any 
other  position  M1  by  a  translation  combined  with  a  rotation  about 
an  axis  parallel  to  the  direction  of  translation,  and  this  can  be 
done  in  only  one  way.  The  axis  so  determined  is  called  the 
central  axis  of  the  displacement. 

The  order  of  translation  and  rotation  about  the  central  axis  is 
indifferent ;  indeed,  translation  and  rotation  might  take  place 
simultaneously. 

41.  A  motion  of  a  rigid  body  consisting  of  a  rotation  about 
an  axis  combined  with  a  translation  parallel  to  the  axis  is  called 
a  screw  motion,  or  a  twist.     We  have  proved  therefore,  in  Art. 
40,  that  the  most  general  displacement  of  a  rigid  body  can  be 
brought  about  by  a  single  twist. 

42.  To  construct   the  central  axis  and   find  the  translation 
and  angle  of  the  twist  when  the  displacement  is  given  by  the 
positions  A^  BQ,  £70  and  Av  B^  6\  of  three  points  of  the  body, 
we  first  remark  that  the  projection  on  the  central  axis  of  the 
displacement  of  any  point,  say  A^A^  is  equal  to  the  translation 
of  the  twist,  and  hence  the  projections  of  the  displacements  of 
all  points  of  the  body  (such  as  A^A^  BQBly  CQC^)  ar.e  all  equal. 
If  therefore  from  any  point  O  we  draw  lines  OA,  OB,  OC  equal 
and  parallel  to  AQAV  BQBV  CQCV  their  ends  A,  B,  C  will  lie  in 
a  plane  TT  perpendicular  to  the  central  axis,  and  the  perpendicu- 
lar p  dropped  from  O  on  this  plane  TT  will  represent  in  length 
and  direction  the  translation  of  the  twist. 

The  direction  of  the  central  axis  being  thus  determined,  we 
find  its  position  in  space  by  projecting  the  displacements  of  any 
two  of  the  three  given  points,  say  AQAl  and  B^BV  on  the  plane 


44-]  SCREW   MOTION.  23 

TT,  and  finding  the  intersection  of  the  perpendicular  bisectors  of 
these  projections.  This  intersection  is  evidently  a  point  of  the 
central  axis,  and  a  perpendicular  through  it  to  the  plane  TT  will 
give  the  central  axis  in  position. 

43.  In  the  case  of  continuous  motion  there  exists  a  central 
axis  for  every  position  of   the  body;   but  its  position  both  in 
space  and  in  the  body  in  general  varies  in  the  course  of  the 
motion.     The  central  axis  at  any  moment  is  therefore  called  in 
this  case  the  instantaneous  axis. 

44.  The  straight  lines  of  space  which  during  the  progress  of 
the  motion  become  instantaneous  axes  for  the  infinitely  small 
twists  of  the  body  form  a  ruled  surface.     Similarly,  the  lines  of 
the  moving  body  which  in  the  course  of  the  motion  come  to 
coincide  with   these  axes  generate  another  ruled   surface.     In 
.any  given  position  of  the  body  these  two  surfaces  are  in  contact 
along  a  line  (the  instantaneous  axis)  which    is  a  generator  in 
each  of   the  two  surfaces.      The  body  has  an  infinitely  small 
rotation  about  this  line  and  at  the  same  time  slides  along  this 
line  through  an  infinitely  small  distance. 

Thus  the  continuous  motion  of  a  rigid  body  in  the  most  general 
-case  can  be  regarded  as  consisting  of  the  combined  rolling  and 
sliding  of  one  ruled  surface  over  another. 


GEOMETRY   OF   MOTION. 


[45- 


V.    Composition  and  Resolution  of  Displacements. 
i.   TRANSLATIONS;  VECTORS. 

45.  All  the  points  of  a  rigid  body  subjected  to  a  translation 
describe  parallel  and  equal  lines  (Art.  9).  The  translation  of 
the  body  is  therefore  fully  determined  by  the  displacement  A^A^ 

of  any  one  point  A  of  the 
body  (Fig.  12),  and  can  be 
represented  geometrically  by 
AQA1  or  any  line  equal  and 
parallel  to  it,  like  01. 

A  segment  of  a  straight 
line  of  definite  length,  direc- 
tion, and  sense  is  called  a 


C, 


Fig.  12. 


vector.  The  sense  of  the 
vector  (see  Art.  6)  which 
expresses  whether  the  translation  is  to  take  place  from  o  to  i  or 
from  i  to  o,  is  indicated  graphically  by  an  arrow-head,  and  in 
naming  the  vector,  by  the  order  of  the  letters,  01  and  10  being 
vectors  of  opposite  sense. 

46.  Imagine  a  rigid  body  subjected  to  two  successive  trans- 
lations. From  any  point  o  (Fig.  13)  draw  a  vector  01 
representing  the  first  translation,  and  from  its  end  I  a  vector 
12  representing  the  second  transla- 
tion. The  vector  02  will  then  repre- 
sent a  translation  that  would  bring 
the  body  directly  from  its  initial  to 
its  final  position.  This  vector  02  is 
called  the  geometric  sum,  or  the  resul- 
tant, of  the  vectors  01  and  12,  which 
are  called  the  components.  The  oper- 
ation of  combining  the  components  into  a  resultant,  or  of 
finding  the  geometric  sum  of  two  vectors,  is  called  geometric 
addition,  or  composition,  of  vectors. 


Fig.   13. 


$o.]  TRANSLATIONS.  2£ 

47.  The  process  of  geometric  addition  explained  in  Art.  46 
for  the  case  of  two  components  is  readily  extended  to  the  gen- 
eral case  of  n  components.     It  thus  appears  that  the  succession  of 
any  number  of  translations  of  a  rigid  body  has  for  its  resultant 
a  single  translation  whose  vector  is  found  by  geometrically  adding 
the  vectors  of  the  component  translations.     (Compare  Art.  7.) 

48.  The  order  in  which  vectors  are  combined,  or  added,  is 
indifferent  for   the  result.     This    is  directly  apparent   from   a 
figure  in  the  case  of  two  vectors  (Fig.  14). 

For  the  case   of  n  vectors   it  follows  from 

JLA 

the  consideration  that  any  order  of  the  vec- 
tors can  be  obtained  by  repeated  interchanges 
of  two  successive  vectors. 

Geometric  addition  agrees,  therefore,  with 
algebraic  addition  in  being  commutative.  Fig.  14. 

49.  The  vector,  as  the  geometric  symbol  of  a  translation,  has 
length,  direction,  and   sense ;   but  it   is    not    restricted    to   any 
definite  position,  the  same  translation  being  represented  by  all 
equal  and  parallel  vectors.     We  express  this  by  saying  that  two 
vectors  are  equal  if  they  are  of  the  same  length,  direction,  and  sense. 

Translations  are  not  the  only  magnitudes  in  mechanics 
that  can  be  represented  by  vectors.  We  shall  see  later  that 
velocities,  accelerations,  moments  of  couples,  etc.,  can  all  be 
represented  by  vectors  and  are  therefore  compounded  into 
resultants  and  resolved  into  components  by  geometric  addition 
and  subtraction.  In  this  lies  the  importance  of  this  subject 
which  in  its  special  application  to  translations  might  appear  too 
simple  and  self-evident  to  require  extended  presentation. 

The  case  when  the  vectors  represent  concurrent  forces  is 
probably  known  to  the  student  from  elementary  physics  as  the 
"parallelogram  "  or  " polygon  "  of  forces. 

50.  A  translation  may  be  resolved  into  two  or  more  translations 
by  resolving  its  vector  into  components. 


26 


GEOMETRY   OF   MOTION. 


[Si- 


When  the  resultant  translation  and  one  of  its  components  are 
given  by  their  vectors,  the  process  of  finding  the  other  com- 
ponent is  called  geometric  sub- 
traction. It  is  effected,  like 
algebraic  subtraction,  by  re- 
versing the  sense  of  the  com- 
ponent to  be  subtracted,  and 
then  geometrically  adding  it 
to  the  resultant  (Fig.  15). 
In  other  words,  the  geometric 
difference  of  two  vectors  AB 
and  CD  is  found  by  geometri- 


Fig.  15. 


cally  adding  to  AB  a  vector 
equal  but  opposite  to  CD. 
Thus,  in  Fig.  15,  02  is  made  equal  and  parallel  to  AB  ;  21  is 
equal  and  parallel  to  CD  reversed,  that  is  to  DC\  01  is  the 
required  difference. 

51.  The  composition  of  translations  by  geometric  addition  of 
their  vectors  (Art.  47)  holds,  not  for  successive  translations  only, 
but,  owing  to  the  commutative  law  (Art.  48),  for  simultaneous 
translations  as  well.     This  is  easily  seen  by  resolving  the  com- 
ponents into  infinitesimal  parts. 

To  obtain  a  clear  idea  of  two  simultaneous  translations  it  is 
best  to  imagine  the  body  as  having  one  of  these  translations 
with  respect  to  some  other  body,  while  the  latter  itself  is  sub- 
jected to  the  other  translation.  A  man  walking  across  the  deck 
of  a  vessel  in  motion,  an  object  let  fall  in  a  moving  carriage,  a 
spider  running  along  a  branch  swayed  by  the  wind,  are  familiar 
examples. 

52.  This  leads  us  to  the  idea  of  relative  motion. 

Properly  speaking,  all  motion  is  relative;  that  is,  we  can 
conceive  of  the  motion  of  a  body  only  with  regard  to  some  other 
body,  called  the  body  of  reference.  If  the  latter  be  regarded  as 
fixed,  the  motion  of  the  former  is  called  its  absolute  motion. 


55-]  TRANSLATIONS.  2/ 

Thus  in  speaking  of  the  motion  of  a  railway  train,  we  usually 
regard  the  earth  as  fixed  and  can  thus  call  the  displacement  oi 
the  train  from  one  station  to  another  an  absolute  displacement. 
If,  however,  the  motion  of  the  earth  with  regard  to  the  sun 
be  taken  into  account,  the  displacement  of  the  train  from 
station  to  station  is  the  relative  displacement  of  the  train  with 
respect  to  the  earth ;  and  its  absolute  displacement  would  be 
found  by  combining  this  relative  displacement  with  the  abso- 
lute displacement  of  the  earth  (with  respect  to  the  sun  regarded 
as  fixed). 

53.  It  follows  that  when  the  two  displacements  are  transla- 
tions the  absolute  displacement    of  the   body  will  be  found  by 
geometrically  adding   its    relative  displacement   to    the  absolute 
displacement  of  the  body  of  reference.     And  conversely,  the  rela* 
tive  displacement  of  a  body  is  found  by  geometrically  subtracting 
from  its  absolute  displacement  the  absolute  displacement  of  the 
body  of  reference. 

54.  Analytically,  the  composition  and  resolution  of  vectors  is 
merely  a  problem  of  trigonometry.     Thus,  the  resultant  of  two 
sectors  is  the  diagonal  of  the  parallelogram  formed  by  the  two 
vectors  as  adjacent  sides ;  the  resultant  of  three  vectors   is  the 
diagonal  of  the  parallelepiped  having  the  three  vectors  as  con- 
current edges. 

55.  In  the  case  of  more  than  two  or  three  vectors,  however, 
the   solution   by  ordinary  trigonometry  would   become   rather 
tedious,  and  it  is  best  to  proceed  as  follows : 

Assume  an  origin  O  and  three  rectangular  axes  Ox,  Oy,  Oz, 
.and  project  each  vector  on  the  three  axes ;  let  X>  Y,  Z  be  its 
.  projections.  These  projections  X,  Y,  Z  are  three  vectors  whose 
geometrical  sum  is  equal  to  the  vector.  If  n  vectors  were 
originally  given,  we  should  now  have  them  replaced  by  3  n  com- 
ponents of  which  n  lie  in  each  axis.  The  components  lying  in 
the  same  axis  can  be  added  algebraically ;  let  their  respective 


28  GEOMETRY   OF   MOTION.  [56 


sums  be  ^X,  S  F,  ^Z.  The  «  vectors  are  therefore  equivalent  to 
the  three  vectors  ^Xy  ^Yy^Z,  which  form  the  concurrent  edges 
of  a  rectangular  parallelepiped  whose  diagonal  drawn  through 
the  origin  O  is  the  resultant  vector  OR  =  R,  i.e. 


R  = 
The  direction  of  this  vector  is  given  by  the  equations 

,  cos  J3=2,  cos  7=H 


where  a,  /?,  .7  are  the  angles  made  by  OR  with  the  axes  6V,  Oy> 
Oz,  respectively. 

If  all  the  vectors  lie  in  the  same  plane,  we  have  simply  : 


2,  tan  a  = 


56.   Exercises. 

(1)  A  ship  sails  first  5  miles  N.  30°  E.,  then  12  miles  N.  60°  E.,  and 
finally  25   miles  E.   75°  S.     Find  distance  and  bearing  of  the  point 
reached  :   (a)  graphically,  (b)  analytically. 

(2)  Is  a  scale  of  8  miles  to  the  inch  sufficient  to  obtain  the  results  of 
Ex.  ( i)  correctly  to  whole  miles  and  degrees  ? 

(3)  A  rigid  body  undergoes  three  translations,  of  i,  2,  and  3  feet,, 
whose  directions  are  respectively  parallel  to  the  three  sides  of  an  equi- 
lateral triangle  taken  the  same  way  round.     Find  the .  resulting  dis- 
placement. 

(4)  A  ship  is  carried  by  the  current  2  miles  due  W.,  and  at  the  same 
time  by  the  wind  4  miles  due  N.E.,  and  by  her  screw  n  miles  E.  30° 
S.     Find  her  resultant  displacement. 

(5)  A  ferry-boat  crosses  a  river  in  a  direction  inclined  at  an  angle  of 
60°  to  the  direction  of  the  current.     If  the  width  of  the  river  be  half  a 
mile,  what  are  the  component  displacements  of  the  boat  along  the  river 
and  at  right  angles  to  it  ? 

(6)  Two  vectors  of  equal  length  a  are  inclined  to  each  other  at  an 
angle  a.     Find  the  resultant  in  magnitude  and  direction. 

(7)  For  what  angle  a,  in  Ex.  (6),  is  the  resultant  equal  in  magni- 
tude :   (a)  to  each  component  a  ?     (b)  to  J  a  ? 


56.]  TRANSLATIONS.  29 

(8)  Resolve  a  vector  a  into  two  components  making  with  the  vector 
angles  of  30°  and  45°  on  opposite  sides. 

(9)  Steering  his  boat  directly  across  a  river  whose  current  is  due 
west,  a  man  arrives  on  the  opposite  bank  at  a  point  from  which  the 
starting-point  bears  S.E. ;  the  width  of  the  river  being  1200  feet,  how 
far  has  he  rowed  ?    What  is  the  absolute,  and  what  the  relative,  displace- 
ment of  the  boat  ? 

(10)  Assuming  a  raindrop  to  fall  25  feet  in  a  second  in  a  vertical 
direction,  find  in  what  direction  it  appears  to  be  falling  to  a  man  :  (a) 
walking  at  the  rate  of  5  feet  per  second,  (b)  driving  at  the  rate  of  10 
feet  per  second,  (r)  riding  on  a  bicycle  at  25  feet  per  second,  (d)  in  a 
railroad  car  running  60  feet  per  second. 

(n)  Find  in  magnitude  and  direction  the  resultant  of  8  translations 
of  i,  2,  3,  4,  5,  6,  7,  8  feet,  respectively,  each  component  making  an 
angle  of  45°  with  the  preceding  one  :  (a)  graphically,  (b)  analytically. 

(12)  If  a,  b,  c  are  three  vectors  whose  geometric  sum  is  o,  prove* 
that  a/sin  (be)  =^/sin  (ca)  =r/sin  (a-6>). 

(13)  Find  the  resultant  of  two  translations  represented  in  magnitude 
and  direction  by  two  rectangular  chords  of  a  circle  drawn  from  a  point 
on  its  circumference. 

(14)  From  a  point  C  in  the  plane  of  a  circle  whose  centre  is  O, 
draw  two  lines  at  right  angles  to  each  other  so  as  to  intersect  the  circle 
in  Ay  A'  and  B,  B\  respectively.     Show  that  the  resultant  of  the  four 
vectors  CA,  CA',  CB,  CB<  is  equal  to  twice  CO. 

(15)  Prove  that  the  geometric  sum  of  two  vectors  P^P^  PoP2  issuing 
from  the  same  point  P0  passes  through  the  middle  point  G  of  P\P$  and 
has  a  length  =  2  P0G. 

(16)  Prove  that  the  geometric  sum  of  two  vectors  P$P\  and  /o/a  is 
equal  to  (» -f  i)P0G  if  G  be  found  as  follows  :  on  P^  take  Q  so  that 

P0Q  =  -PoPlt  and  on  OP*  take  G  so  that  QG  =  —^—  QP2. 
n  n+  i 

(17)  Show  that  Ex.  (15)    is  a  special  case  of  Ex.  (16). 

(18)  Prove  the  following  rule  for  constructing  the  geometric  sum  of 
n  vectors  P(}P1}  PoPz,  PoPs,  •  •  •  PoPn  issuing  from   the   same   point  P0: 
on  P^   take    Gl  so   that  P^  —  \P^PZ\    on   G&  take    G2  so  that 
G1G2  =  $G1P3;  on  G^  take  G3  so  that  G2G3  =  ±G2P4;  and  so  on. 
If  G  be  the  last  point  so  determined,  the  geometric  sum  of  the  n  vectors 
is  =nP0G. 


30  GEOMETRY   OF   MOTION.  [57. 

2.     ROTATIONS  J    ROTORS. 

57.  When  a  rigid  body  has  a  motion   of   rotation  about  a 
fixed  axis,  all  its  points  with  the  exception  of  those  on  the  axis 
describe  circular  arcs  whose  centres  are  situated  on  the  axis 
(Art.  10). 

The  elements  determining  a  rotary  displacement,  or  a  rotation^ 
are  the  axis  and  the  angle  of  rotation.     These  elements  can  be 
represented   by  a   single   geometrical  symbol ;  we 
have  only  to  lay  off  on  the  axis  of  rotation  a  length 
01  (Fig.  16)  representing  on  some  scale  the  magni- 
tude of  the  angle  6.     An  arrow-head  can  be  used 
to  mark  the  sense  of  the  angle.     It  is  customary, 
at  least  in  English  works  on  mechanics,  to  adopt 
the  counter-clockwise  sense  of  rotation  as  positive. 
The  arrow-head  should  then  be  placed  at  that  end 
Fi     16        of   the  line  representing  the  angle  6  from  which 
the  rotation  appears   counter-clockwise  in  a  plane 
through  the  other  end  at  right  angles  to  the  axis.     The  arrow 
then  points  in  the  direction  in  which  an  ordinary  screw  moves 
when  turned  in  the  positive  sense. 

This  geometrical  symbol  of  a  rotation,  01,  has  been  called  a 
rotor.  It  becomes  of  importance  in  the  case  of  infinitesimal 
rotations,  as  we  shall  see  later  (Art.  68). 

58.  Two  or  more  rotations  about  the  same  axis  can  evidently 
be  combined  into  a  single  rotation  about  the  same  axis  whose 
angle  is  the  algebraic  sum   of   the  angles  of   the   component 
rotations  (Art.  12).     As  regards  rotations  about  different  axes, 
we  have  to  distinguish  three  cases  :  intersecting  axes,  parallel 
axes,  and  crossing  or  skew  axes. 

It  will  be  shown  in  the  following  articles  that  rotations  about 
intersecting  or  parallel  axes  can  always  be  combined  into  a 
single  rotation  which  may  happen  to  reduce  to  a  translation. 

Rotations  about  skew  axes  cannot  in  general  be  reduced  to  a 


59-] 


ROTATIONS. 


single  rotation  or  translation  ;  it  will  be  shown  in  the  next  sec- 
tion (Arts.  74-79)  that  they  reduce  to  a  twist,  or  screw  motion. 

59.  Intersecting  Axes.  The  resultant  of  two  successive  rota- 
tons, 0J  about  \  and  #2  about  12,  when  the  axes  \  and  12  intersect 
in  a  point  O,  is  a  single  rotation  of  angle  6  about  an  axis  1  passing 
through  O.  The  trihedral  formed  by  /x,  /2  and  /  has  at  /x  a  dihe- 
dral angle  =  \  0lt  at  /2  a  dihedral  angle  =  —  \Q^  while  its 
exterior  angle  at  /is  =J#;  that  is,  we  have  on  a  sphere  of 
radius  I  described  about  O  : 


cos       = 


cos 


—  sn 


sn 


cos 


(i) 


sn 


The  truth  of  this  proposition  will  appear  by  considering  Fig.. 
17.  The  rotation  61  about  the  axis  /x  brings  the  axis  /2  into  its- 
final  position  /'2.  The  rotation  0% 
about  l\  brings  /x  into  its  final 
position  l\.  The  planes  bisecting 
the  dihedral  angles  Ol  at  /j  and  02 
at  l\  intersect  in  a  line  /  which  by 
the  rotation  B1  about  l^  is  brought  ' 
into  the  position  I',  and  by  the 
rotation  02  about  /'2  is  brought 
back  into  its  original  position  /. 
The  effect  of  the  two  rotations 
taken  in  this  order  is  therefore  to 
leave  the  line  /  in  its  place  ;  that 
is,  the  resultant  of  the  two  succes- 
sive rotations  is  a  single  rotation 
about  /  as  axis.  Moreover,  inspec- 
tion of  the  figure  shows  that  a 
rotation  about  /  by  an  angle  equal 

to  twice  the  exterior  angle  of  the  trihedral  //x/2  at  /  brings 
and  /2  into  their  final  positions  l\  and  l\. 


Fig.  17. 


GEOMETRY    OF   MOTION. 


[60. 


60.  It  is  to  be  noticed  that  /x  and  /2  are  here  regarded  as 
lines  of  the  rigid  body ;  and  while  /x  coincides  with  the  position 
of  the  first  axis  of  rotation  in  space,  the  second  axis  of  rotation  in 
space  has  the  position  l\,  and  not  /2.     It  follows  that,  in  general, 
the  order  of  the  two  rotations  is  not  indifferent.     But  by  repeat- 
ing the  construction,  any  number  of  rotations  taken  in  a  definite 
order  can  be  combined  into  a  single  rotation  provided  every  axis 
intersects  the  axis  of  the  resultant  of  all  preceding  rotations. 

61.  Again,  in  finding  /  from  /x  and  /2,  the  positions  of  the 
axes  in  the  rigid  body,  as  we  did  in  Art.  59,  the  angle  J^  is  to 
be  applied  to  the  plane  /j/2  at  /j  in  its  proper  sense,  i.e.  on  that 
side  towards  which  the  rotation  about  /x  takes  place  ;  but  ^#2  at 
/2  is  to  be  applied  to  this   plane  in  the  opposite  sense.     If, 
however,  we  wish  to  construct  /  from  the  absolute  positions  of 
the  axes  of  rotation  in  space,  /:  and  /'2,  we  have  to  use  — 
and  +-|02. 

62.  In  the  case  of  two  infinitely  small  rotations,  say  dQ^  and 
d6y    about    intersecting    axes   /x,    /2,    the    construction    gains 
remarkable  simplicity.     The  resulting  axis  /  falls  into  the  plane 

of  the  given  axes. 

Substituting  d6  for  sin#  and 
for   cos#,    the    equations     of    Art.     59 
assume  the  form 


(2'} 


sin  (//2)  _  sin  (/]/2) 
i 


Fig.   18. 
components  d01  and 


These  equations  show  that  dO  can  be 
found  by  geometrically  adding  the 
rotors  (Art.  57)  representing  the  rota- 
tions dO-L  and  dO%.  In  other  words,  the 
s  (or  lengths  proportional  to  them)  being 


laid  off  on  their  respective  axes  (Fig.  18),  the  resultant  rotation 


ROTATIONS. 


33 


dQ  will  be  found  in  magnitude  and  direction  as  the  diagonal  of 
the  parallelogram  whose  adjacent  sides  are  dQ^  and  dO^  just  as 
in  the  case  of  translations  (Art.  46).  The  importance  of  this 
proposition  will  appear  later  (Art.  276). 

It  is  to  be  noticed  that,  in  the  case  of  infinitesimal  rotations, 
the  order  of  succession  in  which  they  take  place  is  obviously 
indifferent;  they  can  therefore  be  imagined  to  take  place 
:simultaneously. 

63.  Parallel  Axes.  The  composition  of  two  successive  rota- 
tions about  parallel  axes  is  not  essentially  different  from  the 
composition  of  rotations  about  two  intersecting  axes.  The 


trihedral  //^  of  Fig.  17,  formed  by  the  given  axes./!,  /2,  and 
the  resulting  axis  /,  becomes  now  a  triangular  prism,  and  the 
spherical  construction  is  replaced  by  a  construction  in  a  plane  at 
right  angles  to  the  axes.  Fig.  19  shows  this  construction  for 
the  case  of  two  rotations  having  the  same  sense  (01  and  #2 
being  of  the  same  sign)  ;  Fig.  20  illustrates  the  case  of  two 
opposite  rotations.  The  letters  have  the  same  meaning  as  in 
Fig.  17. 

PART   I — 3 


34 


GEOMETRY   OF   MOTION. 


[64. 


The  signs  of  Ol  and  #2  being  taken  into  account,  the  formulae 
of  Art.  59  are  now  replaced  by  the  following  : 


sn 


sin  J  Ol     sin  |  6 


The  order  of  two  finite  rotations  about  parallel  axes  is  not 
invertible. 


Fig.  20. 

By  repeating  the. above  construction  it  is  evidently  possible 
to  find  the  resultant  of  any  number  of  successive  rotations 
about  parallel  axes,  the  rotations  being  taken  in  a  definite 
order. 

64.  The  particular  case  of  two  equal  and  opposite  rotations 
about  parallel  axes  deserves  special  consideration.  The  point  L 
lies  at  infinity;  hence,  the  axis  of  rotation  being  at  an  infinite 
distance,  the  resulting  motion  is  a  translation  (Art.  19).  This 
will  also  appear  from  Fig.  21  ;  the  first  rotation,  about  /x,  brings 


66.] 


ROTATIONS. 


35 


the  plane  /j/2  into  the  position 
/'2,  brings  it  into  the  position 
/'/!  which  is  parallel  to  the 
original  position  /^  The 
whole  body  has  thus  been 
moved  parallel  to  itself  in  the 
direction  L^L'^  and  the  mag- 
nitude of  this  translation  is 


\\  the  following  rotation,  about 


Fig.  21. 


.   0 

sin-, 
2 


(3) 


s 


where  0  is  the  angle  of  rotation  about  each  axis,  and  L 
the  distance  of  the  axes. 

The  order  of  the  rotations  is  evidently  not  invertible. 

65.  We  have  seen  in  the  preceding  article  that  two  equal  and 
opposite  rotations  about  parallel  axes  produce  a  translation  at  right 
angles  to  the  axes  of  rotation.    A  translation  can  therefore  always 
be  replaced  by  two  such  rotations.     It  follows  that  a  translation 

followed  by  a  rotation  about  an  axis  at  right  angles  to  the  direc- 
tion of  translation  can  be  replaced  by  a  single  rotation  about  a 
parallel  axis.  To  find  this  resulting  rotation  it  is  only  neces- 
sary to  replace  the  translation  by  two  parallel  equal  and  oppo- 
site rotations  having  the  same  effect  (Art.  64) ;  the  three 
rotations  so  obtained  have  parallel  axes  and  can  therefore 
(Art.  63)  be  combined  into  a  single  one. 

66.  The  case  of   two  infinitely  small  rotations  (Fig.   22)   is 
again  of  particular  importance,  as  we  shall  see  later  on.     The 

formulae    of    Arts.    59    and    63 
become  in  this  case 


k 

i 

Fig.  22. 


J^^-    J  J  -L>  J-^t  f 


(I"') 

(2'") 


The  axis  /  of  the  resulting  rota- 
tion lies  therefore  in  the  plane 


of  the  given  axes  lv  /2  and  divides  their  distance  in  th< 


36  GEOMETRY    OF   MOTION.  [67. 

ratio  of  the  angles  of  rotation.  The  sense  of  the  segments 
L^L,  LL2,  L^L^  must  be  taken  into  account  as  well  as  the 
sense  of  the  angles  dOlt  d92,  dO.  The  axis  /  lies  between  /x 
and  /2  if  dOly  d02  have  the  same  sense ;  otherwise  it  lies  outside 
the  space  between  /x,  /2  on  the  side  of  the  axis  having  the 
greater  angle. 

67.  Two  equal  and  opposite  infinitely  small  rotations  about 
parallel  axes  produce  an  infinitely  small  translation  equal  to 
Z:Z2  •  dd  (see  Art.  64,  Formula  (3) )   directed  at  right   angles 
to  the  plane  of  the  axes  llt  /2.      Conversely,  an  infinitely  small 
translation  can  always  be  replaced  by  two  equal  and  opposite 
infinitesimal  rotations. 

68.  An  infinitesimal  rotation  of   angle  dO  about  an  axis  / 
can   be   represented   (Art.    57)    by  a  rectilinear   segment    laid 
off  on  /  equal  to  d9,  or,  to  avoid    infinitesimal  lengths,   pro- 
portional to  dO.      This  geometrical  symbol  of  an  infinitesimal 
rotation  has  all  the  characteristics  of  a  vector  (compare  Arts.  45, 
49) ;  but  it  has  one  more  which  distinguishes  it  from  the  vector 
representing   a   translation :    it   is   localized,  or   attached   to   a 
definite  line  ;  for  two  equal  and  parallel  rotations  about  different 
axes  do  not  represent  the  same  thing.     Such  a  localized  vector 
is  called  a  rotor. 

69.  The  theory  of   rotors  is  of  just  as  great  importance  in 
mechanics  as  that   of   vectors    (Art.   49).      Angular  velocities, 
momenta,  forces,  all  have  for  their  geometrical  representatives 
rotors,   i.e.  rectilinear   segments    of   definite  direction,    length, 
sense,  and  situated  on  a  definite  line. 

The  theory  of  the  composition  and  resolution  of  rotors  is  a 
matter  of  pure  geometry ;  it  remains  the  same  whatever  the 
rotor  may  represent.  Thus  we  have  seen  in  Art.  62,  in  the  case 
of  infinitesimal  rotations,  that  concurrent  rotors  are  combined  by 
geometrical  addition.  The  same  rule  holds  for  angular  velocities, 
momenta,  and  forces.  In  Art.  66  the  rule  for  combining  two 


7i  ]  ROTATIONS.  37 

parallel  rotors  is  explained  by  the  example  of  infinitesimal  rota- 
tions. The  student  acquainted  with  elementary  physics  will 
recognize  in  this  rule  the  so-called  principle  of  the  lever  which 
is  based  on  the  composition  of  parallel  forces. 

70.  Two  rotors  of  equal  length  and  opposite  sense  situated  on 
parallel  lines  (Fig.  23)  are  said  to  form  a  couple.     The  two  rotors 
Py  P  are  called  the  sides,  their   perpen- 

dicular distance  /  the  arm,  and  the 
product  Pp  the  moment  of  the  couple. 
It  has  been  proved  in  Art.  67  that  a 
couple  of  infinitesimal  rotations  pro- 
duces an  infinitesimal  translation.  In 

general,  a  rotor  couple  is  equivalent  to  a 

,    n          i 
vector,  as  we  shall  see  later. 

71.  The  converse  proposition  of  Art.  67,  viz.  that  an  infini- 
tesimal  translation   can    always   be   replaced    by   a    couple  of 
infinitesimal  rotations,  requires  a  little  further  consideration. 

Suppose  we  wish  to  replace  the  translation  ds  by  a  couple. 
According  to  Art.  67,  the  axes  /lf  /2  of  the  two  rotations  must 
be  at  right  angles  to  ds  ;  the  distance  L^L^  of  the  axes  and  the 
angle  of  rotation  ad  are  only  subject  to  the  condition  that  their 
product  should  equal  ds,  i.e. 


There  is,  therefore,  an  infinite  number  of  couples  equivalent  to 
ds,  all  having  the  same  moment  L^L^  •  ds  and  all  lying  in  a  plane 
perpendicular  to  ds. 

It  thus  appears  that  the  characteristics  of  a  couple  are  its 
moment  and  the  aspect  of  its  plane  ;  in  other  words,  a  couple 
(P,  p)  is  equivalent  to  any  couple  (P1  ',  /')  provided  (a)  that  they 
lie  in  parallel  planes  or  in  the  same  plane,  and  (b)  that  their 
moments  are  equal,  i.e.  />•/  =  /".•/'.  This  allows  us  to  repre- 
sent a  rotor  couple  (P,  p)  by  a  vector  perpendicular  to  the  plane  of 
the  couple  and  equal  in  magnitude  to  its  moment  Pp. 


GEOMETRY   OF   MOTION. 


[72. 


The  sense  of  the  vector  is  determined  as  follows.     In  the  case 
of  infinitesimal  rotations  it  appears  from  Arts.  67  and  64  that  a 

couple  of  the  type  A,  Fig.  24, 
produces  a  translation  upwards 
from  the  plane  of  the  figure, 
i.e.  towards  the  reader  ;  while  a 
couple  of  the  type  B  produces 
a  downward  translation,  away 
from  the  reader.  Regarding  the  couples  as  rigid  figures,  their 
rotors  as  forces,  and  the  middle  point  of  their  arms  as  fixed, 
the  type  A  tends  to  produce  rotation  in  the  counter-clockwise, 
positive,  sense  ;  the  type  B  in  the  negative  sense.  The  former 
is  therefore  regarded  as  positive,  and  its  vector  is  drawn  from 
its  plane  towards  the  reader. 


Fig.  24. 


72.    Let  us  now  return  to  our  infinitesimal  displacements. 

An  infinitesimal  translation  ds  can  be  combined  with  an  infini- 
tesimal rotation  d#  about  an  axis  1  at  right  angles  to  ds  (Fig.  25). 
To  find  the  resultant  single  rotation  we  have  only  to  replace 
the  translation  ds  by  an  equivalent  couple 
whose  angle  of  rotation  we  select  equal  to 
that  of  the  given  rotation ;  that  is,  we  put 
ds  =  L,L»-dd,  whence 


LI  -ds 

L-' 


d6 


-d8 


The  plane  of  the  couple,  being  perpen- 
dicular to  ds,  can  be  taken  so  as  to  contain 
the  axis  /  of  the  given  rotation  dO ;  and  in 
this  plane  the  couple  can  be  so  placed  that  pig  25. 

one  of  its  sides  (see  Fig.  25)  falls  into  this 
axis  /.    Selecting  the  proper  side  of  the  couple,  we  shall  have  on 
/two  equal  and  opposite  rotations  d6,—d0,  which  destroy  each 
other,  leaving  only  the  rotation  d6,  about  an  axis  at  the  distance 
from  /. 


750  TWISTS.  39 

Thus  it  is  seen  that  the  combination  of  an  infinitely  small 
rotation  dO,  with  an  infinitely  small  translation  ds  at  right  angles 
to  the  axis  of  rotation,  produces  a  single  rotation  of  the  same 
angle  about  a  parallel  axis  at  a  distance  ds/d#  from  the  original 
axis  in  the  plane  through  this  axis  perpendicular  to  the  direction 
of  translation. 

73.  Exercises. 

(1)  The  telescope  of  a  theodolite,  originally  horizontal  and  pointing 
north,  is  tipped  into  an  elevation  of  60°,  and  then  turned  into  the  prime 
vertical  so  as  to  point  west.     What  single  rotation  is  equivalent  to  the 
two  successive  rotations? 

(2)  In  the  preceding  example,  what  would  be  the  result  of  inverting 
the  order  of  the  two  rotations  ? 

(3)  The  motion  of  a  man  in  walking  may  be  approximately  described 
as  consisting  at  every  step  of  two  rotations  of  the  body  about  parallel 
axes  perpendicular  to  the  direction  of  motion,  one  axis  passing  through 
the  hip-joint,  the  other  through  the  foot  that  remains  on  the  ground 
while  the   other   foot  is  thrown   forward.      Find   the  angle   of  swing 
(assuming  the  two  rotations  to  be  equal  and  opposite)  if  the  length 
of  the  step  is  15  inches  and  the  height  of  the  hip-joint  3^  feet. 

3.     SCREW    MOTIONS  ;    TWISTS. 

74.  We  have  seen  in  Arts.  40,  41  that  a  twist,  i.e.  a  rota- 
tion combined  with  a  translation  parallel  to  the  axis  of  rotation, 
constitutes  the  most  general  form   of   the   displacement  of  a 
rigid  body.     We  proceed  to  discuss  the  most  important  cases 
of  the  compositions  of  rotations  and  translations  resulting  in 
twists. 

75.  A  rotation  of  angle  6  about  an  axis  /  can  be  combined 
with  a  translation  whose  vector  is  s,  by  resolving  s  into  two 
components ;  sl  perpendicular  to  /,  and  s2  parallel  to  /.     The 
former  component  combines  (by  Art.  65)  with  the  rotation  into 
a  single  rotation  of  the  same  angle  6  about  an  axis  parallel  to  /. 
The  result  is  therefore  a  rotation  accompanied  by  a  translation 
s%  parallel  to  the  axis  of  rotation,  i.e.  a  twist. 


GEOMETRY    OF   MOTION. 


[76. 


76.  If  the  rotation  dO  and  the  translation  ds  are  infinitesimal, 
the  axis  of  the  resulting  twist  has  (by  Art.  68)  a  distance  ds/dQ 
from  the  axis  /  of  the  rotation  dd  and  lies  in  the  plane  laid 
through  /  at  right  angles  to  ds. 

77.  Skew  Axes.     The  resultant  of  two  successive  rotations  6± 
and  02  about  two  skew  axes  \  and  12  is  a  twist.     This  follows  of 

course  from  the  proposition  of 
Art.  40.  The  axis  of  the  re- 
sulting twist  is  the  central  axis  \ 
of  the  displacement ;  its  direc- 
tion and  position  can  be  found 
as  in  Art.  42.  Fig.  26  illus- 
trates the  process.  L^L^  is  the 
shortest  distance  of  the  axes 
/!,  /2.  The  first  rotation,  Ov" 
about  /j,  brings  /2  into  its  final 
position  /'2,  and  L^  into  Lf2 ',  the 
second  rotation,  02  about  /'2, 
brings  /x  into  its  final  position 
/\,  and  L  into  L\.  The  axis 
/  of  the  resulting  twist  will 
evidently  be  the  shortest  dis- 
tance of  the  bisectors  of  the  angles  Z2£1Z'2  and  L^L\L\. 
For  a  rotation  about  this  line  /  brings  /2  into  /'2  and  /x  into  l\. 

78.  The  angle  of  the  resulting  twist  is  the  same  as  the  angle 
of  the  rotation  resulting  from  two  rotations  0V  #2  about  two 
intersecting  axes  parallel  to  the  given  axes  /1?  /2.     For  (by  Art. 
65)  either  one  of  the  rotations,  say  02  about  /2,  may  be  replaced 
by  a  rotation  of  the  same  angle  02  about  an  axis  parallel  to  /3 
and  intersecting  llt  combined  with  a  translation  at  right  angles 
to  /2.     The  two  rotations  about  the  intersecting  axes  can  then 
be  combined  into  a  single  rotation,  and  the  angle  and  direction 
of  the  axis  of  this  latter  rotation  are  not  changed  by  combi- 
nation with  the  translation  (Art.  74). 


Fig.  26. 


8o.]  TWISTS.  4I 

79.  It  follows  from  the  two  preceding  articles  that  a  twist 
can  always  be  resolved  into  two  rotations  about  skew  axes,  and 
this  can  be  done  in  an  infinite  number  of  ways.     It  is  also  easy 
to  see  that  two,  or  any  number  of,  successive  twists  can  be  com- 
bined into  a  single  twist  by  resolving  each  twist  into  its  rotation 
and  translation,  and   combining  all  rotations   into  a  resulting 
twist  and  all  translations  into  a  resulting  translation  ;  the  result- 
ing twist    combined  with  the  resulting  translation   gives   the 
twist  equivalent  to  all  the  given  twists. 

80.  For  a  more  complete  account  of  the  geometry  of  motion  the 
student  is  referred  to  A.  SCHOENFLIES,  Geometric  der  Bewegung,  Leipzig, 
Teubner,   1886;    and  to  W.   SCHELL,   Theorie  der  Bewegung  und  der 
Krafte,  Leipzig,  Teubner,  Vol.  I.,  1879,  pp.  144-187.     See  also  R.  S. 
BALL,  Theory  of  screws,  Dublin,  Hodges,   1876;    and   H.  GRAVELIUS, 
Ball's  theoretische  Mechanik  starrer  Systeme,  Berlin,  Reimer,  1889, — . 
for  the  more  advanced  parts  of  the  subject.     Many  authors  treat  the 
geometry  of  motion  in  connection  with  Kinematics ;  see  the  references 
in  Chapter  H.,  in  particular  the  works  of  Burmester,  Resal,  Villie\ 

Applications  to  mechanism  and  machinery  will  be  found  in  F. 
REULEAUX,  Kinematics  of  machinery,  edited  by  A.  B.  W.  Kennedy,. 
London,  Macmillan,  1876;  in  J.  H.  COTTERILL,  Applied  mechanics, 
London,  Macmillan,  1884,  pp.  99-134;  and  in  ALEX.  B.  W.  KENNEDY, 
The  mechanics  of  machinery,  London,  Macmillan,  1886. 


42  KINEMATICS.  [81. 


CHAPTER    II. 

KINEMATICS. 

I.    Time. 

81.  Before  introducing  the  idea  of  time  into  the  study  of 
motion,  a  word  must  be  said  on  the  measurement  of  time. 

It  is  the  province  of  astronomy  to  devise  methods  for  measur- 
ing time ;  the  usual  method  consists  in  transit  observations. 
Thus  the  fundamental  unit  of  time  in  astronomy,  or  the  sidereal 
day,  is  the  interval  between  two  successive  upper  transits  of  the' 
true  vernal  equinox  over  the  same  meridian. 

82.  For  the  purposes  of  every-day  life,  it  is  more  convenient 
to  make  the  measurement  of  time  depend  on  the  apparent  revo- 
lution of  the  sun.     But  the  interval  between  two  successive 
upper  transits  of  the  sun  over  the  same  meridian,  which  is  the 
true,  or  apparent  solar  day,  is  not  constant  throughout  the  year, 
owing  to  the  inclination  of  the  earth's  axis  to  the  plane  of  its 
orbit  and  to  the  ellipticity  of  this  orbit.     The  true  solar  day  is 
thus  not  well  adapted  to  serve  as  a  unit  of  time. 

Astronomers  imagine,  therefore,  a  so-called  first  mean  sun 
moving  uniformly  in  the  ecliptic  so  as  to  pass  the  perigee  simul- 
taneously with  the  real  sun  ;  and  a  second  mean  sun  moving 
uniformly  in  the  equator  so  as  to  pass  the  vernal  equinox  simul- 
taneously with  the  first  mean  sun.  The  interval  between  two 
successive  passages  of  the  second  mean  sun  over  the  same 
meridian  is  called  the  mean  solar  day.  This  may  be  regarded 
-as  the  standard  on  which  all  time-determinations  in  mechanics 
are  based. 

The   mean   solar  day  is   subdivided   into   24   hours  =  1440 


86.]  TIME. 


43 


minutes  =  86  400  seconds.     In  theoretical  mechanics  the  second 
is  generally  used  as  the  unit  of  time. 

83.  To  reduce  mean  time  to  apparent  time,  it  is  only  neces- 
sary to  subtract  from  mean  time  the  so-called  equation  of  time, 
whose  value  for  any  particular  date  is  given  in  the  Ephemeris. 

84.  The  relation  between  mean  solar  time  and  sidereal  time 
is  readily  found  by  considering  that  the  tropical  year,  i.e.  the 
interval  between  two  successive  passages  of  the  sun  through 
the  mean  vernal  equinox,  has  365.2422  mean  solar  days,  and 
of   course   just   one   more   sidereal    day.     Hence    i    solar  day 
=  366.2422/365.2422  =  1.002738  sidereal  day;   in  other  words, 
the  sidereal  day  contains  86  164.1  seconds  of  mean  time,  while 
the  solar  day  contains  86  400  such  seconds.* 

85.  It  will   have   been   noticed   that   all   these   methods   of 
measuring  time  are  ultimately  based  on  the  assumption  that 
the  rotation  of  the  earth  on  its  axis  is  perfectly  uniform.     Obser- 
vation shows  this  assumption  to  be  true,  or  at  least  to  have  a 
very  high  degree  of  approximation. 

It  might  be  asked  how  we  can  know,  without  using  some  unit  of  time 
for  comparison,  that  the  earth's  rotation  on  its  axis  is  uniform ;  in  other 
words,  that  the  mean  solar  day  is  constant.  Our  absolute  unit  of  time 
would  seem  to  be  obtained  by  reasoning  in  a  circle.  This  objection  is 
not  quite  without  foundation ;  and  as  similar  difficulties  arise  in  the 
case  of  other  fundamental  data  of  mechanics,  it  may  be  well  to  consider 
the  matter  a  little  more  in  detail. 

86.  The  simplest  answer  is  that  we  assume  the  constancy  of  the 
mean  solar  day  and  find  this  assumption  fully  justified  by  the  fact  that 
while  the  whole  structure  of  the  astronomical  and  physical  sciences  rests 
on  this  assumption,  the  theoretical  predictions  of  these  sciences  are 
found  to  be  in  close  agreement  with  the  results  of  direct  observation. 

Historically,  the  assumption  was  originally  adopted  on  account  of  its 

*  For  further  particulars  see  W.  CHAUVENET,  Spherical  and  practical  astronomy, 
Vol.  I.,  p.  52  sq.  and  pp.  651-654;  also  the  American  Ephemeris  and  Nautical 
Almanac. 


44  KINEMATICS.  [87. 


simplicity,  as  a  practical  working  hypothesis,  and  it  was  found  to  work 
well.  From  the  logical  point  of  view  we  may  strengthen  its  probability 
by  the  following  considerations. 


87.  The  origin  of  our  notion  of  time  as  a  measurable  quantity  lie 
in  the  subjective  sensation  that  teaches  us  instinctively  to  distinguish 
between  shorter  and  longer  intervals  of  time.  This  feeling  of  time  is 
of  course  (just  as  in  the  analogous  case  of  muscular  force)  far  too  vague 
and  indefinite  to  admit  of  measurement.  But  it  is  sufficient  to  convince 
us  that,  approximately,  the  lengths  of  successive  days  are  equal.  With? 
far  greater  approximation  can  we  judge  by  our  time-feeling  that  the 
oscillations  of  the  pendulum  of  a  clock  are  nearly  isochronous.  Let  us 
combine  these  two  entirely  independent  facts.  Careful  observation  will 
show  that  the  number  of  oscillations  made  by  the  pendulum  in  the 
interval  between  two  culminations  of  the  mean  sun  is  almost  precisely 
the  same  for  every  mean  day.  Moreover,  the  agreement  becomes  the 
more  perfect  the  more  we  eliminate  any  causes  that  tend  to  disturb 
the  isochronism  of  the  pendulum.  It  will  therefore  be  reasonable  to 
conclude  that  the  mean  solar  day  must  have  a  very  nearly  constant 
length. 

But  it  is  to  be  kept  in  mind  that  this  is  an  empirical  fact  and  hence 
not  absolutely  true,  but  only  within  the  limits  of  the  errors  of  observa- 
tion. Indeed,  certain  considerations  concerning  the  friction  caused 
by  the  tides  make  it  probable  that  the  angular  velocity  of  the  earth  is 
diminishing  very  slowly.* 

*  See  O.  RAUSENBERGER,  Analytische  Mechanik,  I.,  Leipzig,  Teubner,  1888,  p.  14;  i 
H.  STREINTZ,  Physikalische  Grundlagen  der  Mechanik,  Leipzig,  Teubner,  1883,  p. 
8 1  sq.;  E.  BUDDE,  Allgemeine  Mechanik,  I.,  Berlin,  Reimer,  1890,  p.  33;  THOMSON 
and  TAIT,  Natural  philosophy,  I.,  London,  Macmillan,  1879,  p.  460;  J.  C.  MAXWELL, 
Matter  and  motion,  New  York,  Van  Nostrand,  1878,  p.  27  and  p.  60.;  K.  PEARSON,. 
Grammar  of  science,  London,  Scott,  1892,  pp.  217-^230. 


: 


9i.]  VELOCITY.  45 

II.     Linear  Kinematics. 

I.     UNIFORM    RECTILINEAR    MOTION;    VELOCITY. 

88.  Consider  a  point  moving  in  a  straight  line.     If  through- 
out the  whole  motion  equal  spaces  are  always  described  in  equal 
times,  the  motion  is  said  to  be  uniform. 

89.  Next   consider  two  points  each  moving  uniformly  in  a 
straight  line.     The  motions  may  still  be  different  ;  for  it  is  pos- 
sible that  while  one  of  the  points  moves  in  a  given  time  t  over  a 
space  sv  the  other  moves  during  the  same  time  /  over  a  different 
space  s2.     The  points  are  then  said  to  have  different  velocities, 
.and  their  velocities  are  said  to  be  as  sl  is  to  s2.     The  velocity  v 
of  uniform  motion  is  therefore  measured  by  the  ratio  of  the 
.space  s  described  in  any  time  t  to  this  time  ;  that  is,  v=s/t. 

90.  This  equation  written  in  the  form 

s  =  vt  (i) 

is  called  the  equation  of  motion  of  the  point.     It  follows  from 
Art.  89  that  in  uniform  motion  the  velocity  v  is  constant. 

With  t  as  abscissa  and  s  as  ordinate  (or  vice  versa],  the  equa- 
tion of  uniform  motion  (i)  represents  a  straight  line  ;  the 
tangent  of  the  angle  made  by  this  line  with  the  axis  of  /  repre- 
sents the  velocity  v. 

91.  Let  the  point  P  start  at  the  time  t=o  from  a  point  O 
(Fig.  27);  let  it  reach  the  point  PQ  at  the  time  t=tQ  and  the 


Fig.  27. 

point  Pl  at  the  time  t=t.  Then,  putting  OPQ=sQ)  OPl  =  s,  the 
space  passed  over  in  the  time  t—  tQ  is  s—  J0  ;  hence  the  velocity 
•v=(s  —  s^)/(t—  t^.  The  equation  of  uniform  motion  can  there- 
fore be  written  in  the  form 


46  KINEMATICS.  [92. 

If  the  times  be  counted  from  the  instant  when  the  moving 
point  is  at  PQ,  we  have  /0=o,  and  the  equation  of  motion  is 


Finally,  if  both  times  and  spaces  are  counted  from  PQ  as 
origin,  we  have  J0  =  o,  so  that  (i")  reduces  to  (i). 

92.    To  measure  velocities  we  must  adopt  a  unit  of  velocity. 

In  kinematics,  the  only  fundamental,  i.e.  independent,  units 
required  are  those  of  length  and  time.  All  other  quantities 
can  be  expressed  in  terms  of  length  and  time,  and  their  units 
are  therefore  called  derived  units. 

Thus,  the  definition  of  the  velocity  of  uniform  motion  as  a 
length  divided  by  a  time  (Art.  89)  can  be  expressed  by  the 
symbolic  equation 


and  we  say  that  the  dimensions  of  velocity  are  I  in  length  and 
—  i  in  time. 

When  L  =  l  and  T  =  l,  we  have  V  =  l.  We  must  therefore 
select  for  our  unit  of  velocity  that  velocity  with  which  unit 
length  is  described  in  unit  time. 

Hence  in  the  C.  G.  S.  system  (see  Arts.  13,  14)  the  unit 
velocity  is  a  velocity  of  i  cm.  per  second  ;  in  the  F.  P.  S.  system 
it  is  a  velocity  of  i  ft.  per  second. 

93.  In  practice  other  units  are  often  used,  and  the  same 
concrete  velocity  can  therefore  be  expressed  by  different  num- 
bers. Thus  the  same  velocity  of  a  railroad  train  can  be 
described  as  30  miles  per  hour,  or  44  ft.  per  second,  or  (approx- 
imately) 13.41  metres  per  second. 

The  symbols  s,  v,  t,  etc.,  in  the  kinematical  equations  must  be 
understood  to  represent  the  numerical  ratios  of  the  concrete 
quantities  to  their  respective  units.  The  symbol  v,  for  instance, 
stands  for  the  ratio  V/Vi  of  the  concrete  velocity  Fto  its  unit 


96.]  VELOCITY.  47 

p  and  we  have  of  course  the  proportion  :  30  miles  an  hour  is  to 
I  mile  an  hour  as  44  ft.  per  second  is  to  I  ft.  per  second,  etc. 

94.  The  full  meaning  of  the  equation  of  dimensions  V  =  LT~1 
is  obtained  if  we  substitute  V/V^  for  V,  L/Ll  for  I,  T/T^  for'T, 
where  V,  L,  T  are  the  concrete  quantities  and  Vlt  Llt  7\  their 
units.     We  find 

Z  -L.   !i 

V,~  L,'  T^ 

and  this  equation  shows  two  things  which  are  of  frequent  appli- 
cation in  reductions  between  different  systems  of  units  : 

(a)  The  numerical  value  V/V^Q/l  a  velocity  varies  directly  as 
the  unit  of  time  and  inversely  as  the  unit  of  length  ; 

(b)  the  unit  of  velocity    V^  varies   directly  as  the  unit    of 
length  and  inversely  as  the  unit  of  time.* 

95.  In    speaking   of   velocities,  the    time    unit    (usually   the 
second)    is   frequently    understood    without    being   mentioned. 
This  has  led  to  considering  velocity  as  a  length  (viz.  the  length 
passed  over  in  unit  time) ;  it  can  then  be  represented  graphi- 
cally by  a  segment  of  a  straight  line,  and  if  in  addition  we,  com- 
bine with  the  idea  of  velocity  that  of  the  direction  and  sense  of 
the  motion,  its  geometrical  representative  will  be  a  vector  (see 
Art.  45).     We  shall  see  later  that    this  view  is  of  particular 
advantage  in  studying  the  velocity  of  curvilinear  motion. 

Some  recent  writers  on  mechanics  use  the  term  velocity 
exclusively  in  this  meaning,  i.e.  as  denoting  a  vector,  and  apply 
the  term  speed  to  denote  the  numerical  magnitude  of  this 
vector.  In  linear  kinematics  the  direction  is  given,  and  the 
"speed"  alone  is  the  subject  of  investigation.  The  +  or  — 
sign  of  the  "speed  "  expresses  the  sense  of  the  motion.f 

96.  Exercises. 

(i)    A  train  leaves  the  station  A  at  9  h.  5  m.,  passes  (without  stop- 

*See  J.  D.  EVERETT,  C.  G.  S.  system  of  units,  1891,  p.  3. 

f  See  Syllabus  of  elementary  dynamics,  Part  I.,  prepared  by  the  Association  for 
the  Improvement  of  Geometrical  Teaching;  London,  Macmillan,  1890,  p.  8. 


48  KINEMATICS.  [96. 

ping)  B  at  9  h.  31  m.,  C  at  9  h.  47  m.,  and  arrives  at  D  at  9  h.  59  m., 
the  distance  AD  being  36.9  miles.     Considering  the  motion  as  uniform  : 

(a)    What  is  the  velocity? 

(£)    What  is  the  equation  of  motion? 

(c)  What  are  the  distances  BD  and  CD? 

(d)  If  after  stopping  5  minutes  at  D  the  train  goes  on  with  the  same| 
velocity  as  before,  when  will  it  reach  £,  loj  miles  beyond  D? 

(e)  Construct  a  graphical  time-table,  taking  the  times  as  abscissas  and  ] 
the  distances  as  ordi nates. 

(2)  Interpret  equations  (if)  and  (i")  geometrically. 

(3)  A  train  leaves  Detroit  at  9  h.  5  m.  A.M.  and  reaches  Chicago . 
at  4  h.  30  m.  P.M.  ;   another  train  leaves  Chicago  at  1 2  h.  20  m.  and 
arrives  in  Detroit  at  7  h.  25  m.  P.M.      The  distance  is  285  miles.     Re-] 
garding  the  motion  as  uniform  and  neglecting  the  stops,  find,  both] 
analytically  and  graphically,  when  and  where  the  trains  will  meet. 

(4)  Reduce  the  following  velocities  to  F.  P.  S.  units  :   (a)  Walking  4  \ 
miles  an  hour;   (3)  trotting  a  mile  in  2  m.  10  s. ;    (c)  railroad  train,] 
from  30  to  50  miles  per  hour;  (^/)  bicyclist,  2  miles  in  4  m.  59^  s. ; 
(f)  sound  in  air,  333  metres  per  second. 

(5)  What  is  the  numerical  value  of  a  velocity  of  22  ft.  per  second 
when  the  hour  is  taken  as  unit  of  time  and  the  mile  as  the  unit  of 
length? 

(6)  How  is  the  unit  of  velocity  changed  if  the  minute  be  adopted  as 
unit  of  time,  the  unit  of  length  remaining  unchanged  ? 

(7)  The  mean  distance  of  the  sun  being  92  J  million  miles,  what  is  the 
velocity  of  light  if  it  takes  light  16  m.  40  s.  to  cross  the  earth's  orbit? 

(8)  Two  trains  are  running  on  the  same  track  at  the  rate  of  25  and 
15  miles  per  hour,  respectively.     If  at  a  certain  instant   they  are   10 
miles  apart,  find  when  they  will  collide  (a)  if  they  are  headed  the  same 
way ;  (6)  if  they  run  in  opposite  directions. 

(9)  In  what  latitude  is  a  bullet  shot  west  with  a  velocity  of  1320  ft. 
per  second  at  rest  relatively  to  the  earth's  axis,  the  radius  being  taken 
as  4000  miles  ? 

(10)  Two  trains,  one  250,  the  other  440  ft.  long,  pass  each  other  on 
parallel  tracks  in  opposite  directions  with  equal  velocity.     A  passenger 
in  the  shorter  train  observes  that  it  takes  the  longer  train  just  4  seconds 
to  pass  him.     What  is  the  velocity? 


ACCELERATION. 


101.  If  v  be  given  as  function  of  /,  say  ^  =  (/>(/),  we  find  from 
(2)  ds  =  vdt,  and  hence  by  inte- 
gration 

j-j0=j[W/,  (3) 

where  SQ  is  the  space  de- 
scribed during  the  time  /0. 
The  equation  v  =  <f>(t}  furnishes 
a  graphical  representation  of  Fig  29. 

the  velocity,   and   formula    (3) 

shows  that  the  space  s  —  SQ  described  during  the  time  t—tQ  is 
represented  by  the  area  included  between  the  curve  v  =  <f>(t),  the 
axis  Ott  and  the  ordinates  ^0  and  v  corresponding  to  /0  and  /, 
respectively  (Fig.  29). 

102.  Similarly,  if  v  be  given  as  a  function  of  s,  say  v  =  \lr(s), 
we  have  from  (2)  dt=ds/v,  and  hence 

(4) 

The  two  velocity  curves  v  =  <f>(t}  and  v  —  ^r(s)  are  of  course  in 
general  different,  and  must  not  be  confounded  with  the  path  of 
the  moving  point,  which  is  here  supposed  rectilinear. 

103.  We  have  seen  (Art.  91,  equation  (i"))  that  in  the  case  of 
uniform  motion  the  velocity  v=(s  —  s^/t,  i.e.  the  rate  of  change 
of  space  with  time,  is  constant.     The  simplest  case  of  variable 
motion  is  that  in  which  the  velocity  varies  uniformly.      The  rate 
at  which  the  velocity  varies  ivith  the  time  is  called  the  accelera- 
tion;  we  shall  denote  it  by/. 

If  the  velocity  vary  uniformly,  the  acceleration  is  constant,  and 
we  have  j=(v  —  v^/tt  where  /  is  the  time  during  which  the 
velocity  changes  from  VQ  to  v. 

By  reasoning  analogous  to  that  employed  in  Art.  99,  we  find 
for  the  acceleration  of  any  rectilinear  motion  at  the  time  t 

%  =  %;•  (s) 


52  KINEMATICS.  [104. 

that  is,  in  rectilinear  motion  the  acceleration  at  any  point  or 
instant  is  the  value,  at  that  point  or  instant,  of  the  second  deriva- 
tive of  the  space  with  respect  to  the  time. 

Negative  acceleration  will  thus  indicate  a  decreasing  veloc- 
ity. 

104.  When  the  acceleration  is  constant,  the  motion  is  said  to 
be  uniformly  accelerated.     In  the  case  of  variable  acceleration 
we  might  again  consider  its  rate  of  change,  which  may  be  called 
the   acceleration   of  the   second  order ;   and  so  on.      Compare 
Art.  156. 

105.  Conformably  to  the  definition  of  acceleration,  its  unit  is 
the  "cm.  per  second  per  second"  in  the  C.  G.  S.  system,  and 
the  "foot  per  second  per  second"  in  the  F.  P.  S.  system.     As 
it  can  rarely  be  convenient  to  use  two  different  time  units  in  the 
unit  of  acceleration  (say,  for  instance,  mile  per  hour  per  second), 
it  is  customary  to  mention  the  time  unit  but  once  and  to  speak 
of  an  acceleration  of  so  many  feet  per  second,  or  cm.  per  sec- 
ond, it  being  understood  that  the  other  time  unit  is  also  the 
second. 

For  the  dimensions  of  acceleration  we  have  (see  Art.  92) 


Denoting,  as  in  Arts.  93,  94,  the  concrete  value  of  an 
acceleration  by  Jt  its  unit  by  Jlt  and  similarly  for  length  and 
time,  we  have  the  equation 

J__L_  Tf 


which  shows  that  (a)  the  numerical  value  J/J^  of  an  acceleration 
varies  directly  as  the  square  of  the  unit  of  time*  and  inversely 
as  the  unit  of  length  ;  and  (b)  the  unit  of  acceleration,^,  varies 
directly  as  the  unit  of  length,  and  inversely  as  the  square  of  the 
unit  of  time. 


ID;.]  RECTILINEAR  MOTION.  53 

106.   Exercises. 

1 i )  A  point  moving  with  constant  acceleration  gains  at  the  rate  of 
30  miles  an  hour  in  every  minute.     Express  its  acceleration  in  F.  P.  S. 
units. 

(2)  At  a  place  where  the  acceleration  of  gravity  is  g=  9.810  metres 
per  second,  what  is  the  value  of  g  in  feet  per  second  ? 

(3)  A  railroad  train,  10  minutes  after  starting,  attains  a  velocity  of 
45  miles  an  hour;  what  was  its  average  acceleration  during  these  10 
minutes  ? 

(4)  If  the  acceleration  of  gravity,  g=  32  feet  per  second,  be  taken 
as  unit,  what  is  the  acceleration  of  the  railroad  train  in  Ex.  (3)  ? 


3.    APPLICATIONS. 

107.   Uniformly  Accelerated  Motion.     As  in  this  case  the  accel- 
eration/is constant  (see  Art.  103),  the  equation  of  motion  (5) 


can  readily  be  integrated  : 

v=jt+C. 

To  determine  the  constant  of  integration  C,  we  must  know  the 
value  of  the  velocity  at  some  particular  moment  of  time.  Thus, 
if  V  =  VQ  when  /=o,  we  find  vQ=C;  hence,  substituting  this 

value  for  C, 

v  —  vQ=jt.  (6) 

This  equation,  which  agrees  with  the  definition  of  /  given  in 
Art.  103,  gives  the  velocity  at  any  time  t.  Substituting  ds/dt 
for  v  and  integrating  again,  we  find  s  =  vQt+^jfi+C  ,  where  the 
constant  of  integration,  C,  must  again  be  determined  from 
given  "initial  conditions."  Thus,  if  we  know  that  S=SQ  when 
/=o,  we  find  sQ=C'  ;  hence 

(7) 


54  KINEMATICS.  [108. 

This  equation  gives  the  space  or   distance   passed  over  in 
terms  of  the  time. 

108.   Eliminating/  between  (6)  and  (7),  we  obtain  the  relation 


which  shows  that  in  uniformly  accelerated  motion  the  space 
can  be  found  as  if  it  were  described  uniformly  with  the  mean 
velocity  J  ( 


109.  To  obtain  the  velocity  in  terms  of  the  space,  we  have 
only  to  eliminate  t  between  (6)  and  (7)  ;  we  find 

V)  =  /('-'<))•  (8) 

This  relation  can  also  be  derived  by  eliminating  dt  between  the 
differential  equations  v  =  ds/dt,  dv/dt=j,  which  gives  vdv  =  jds, 
and  integrating.  The  same  equation  (8)  is  also  obtained 
directly  from  the  fundamental  equation  of  motion  d2s/dt2=jby 
a  process  very  frequently  used  in  mechanics,  viz.  by  multiplying 
both  members  of  the  equation  by  dsjdt.  This  makes  the  left- 
hand  member  the  exact  derivative  of  \(ds/dt}^  or  |V,  and  the 
integration  can  therefore  be  performed. 

110.  The  three  equations  (6),   (7),  (8)  contain  the  complete 
solution  of  the  problem  of  uniformly  accelerated  motion.     For 
uniformly  retarded  motion,  taking  the  direction  of  motion  as 
positive,  we  have  only  to  write  —  /  for  +/. 

If  the  spaces  be  counted  from  the  position  of  the  moving 
point  at  the  time  t=o,  we  have  ^0=o,  and  the,equations  become 


j  I3-]  RECTILINEAR  MOTION.  55 

111.  If  in  addition  the  initial  velocity  VQ  be  zero,  the  point 
.starting  from  rest  at  the  time  ^=o,  the  equations  reduce  to  the 
following  : 

v=jt,  (6") 

•   ":,.  (7") 

;*          ."  '       (8") 

112.  The  most  important  example  of.  uniformly  accelerated 
motion  is  furnished  by  a  body  falling  in  vacuo  near  the  earth's 
surface.     Assuming  that   the  body  does  not  rotate  during  its 
fall,  its  motion  relative  to  the  earth  is  a  mere  translation,  and 
it  is  sufficient  to  consider  the  motion  of  any  one  point  of  the 
body.     It  is  known  from  observation  and  experiment  that  under 
these  circumstances  the  acceleration  of  a  falling  body  is  con- 
stant at  any  given  place  and  equal  to  about  9^0  cm.,  or  32  ft., 
per  second  per  second.  ;  the  value  of  this  so-called  acceleration 
•of  gravity  is  usually  denoted  by  g. 

In  the  exercises  on  falling  bodies  (Art.  114)  we  make  through- 
out the  following  simplifying  assumptions  :  the  falling  body 
does  not  rotate  ;  the  resistance  of  the  air  is  neglected,  or  the 
body  falls  in  vacuo  ;  the  space  fallen  through  is  so  small  that 
g  may  be  regarded  as  constant  ;  the  earth  is  regarded  as  fixed, 
i.e.  we  consider  only  the  relative  motion  of  the  body  with  respect 
to  the  earth. 

,. 

113.  The  velocity  v  acquired  by  a  falling  body  after  falling 

from  rest  through  a  height  h  is  found  from  (8")  as 


This  is  usually  called  the  velocity  due  to  the  height  (or  head)  h, 

while 


is  called  the  height  (or  head)  due  to  the  velocity  v. 


56  KINEMATICS.  [114. 

114.   Exercises. 

(1)  A  body  falls  from  rest  at  a  place  where  g=  32.2.     Find  (a)  the 
velocity  at  the  end  of  the  third  second  ;   (b)  the  space  fallen  through  in 
5  seconds ;  (c)  the  space  fallen  through  in  the  fifth  second. 

(2)  If  a  railroad  train,  at  the  end  of  2  m.  40  s.  after  leaving  the 
station,  has  acquired  a  velocity  of  30  miles  per  hour,  what  was  its  accel- 
eration (regarded  as  constant)  ? 

(3)  Galilei,  who  first  discovered  the  laws  of  falling  bodies,  expressed 
them  in  the  following  form  :   (# )  The  velocities  acquired  at  the  end  of 
the  successive  seconds  increase  as  the  natural  numbers ;   (^)  the  spaces 
described  during  the  successive  seconds  increase  as  the  odd  numbers ; 

(c)  the  spaces  described  from  the  beginning  of  the  motion  to  the  end 
of  the  successive  seconds  increase  as  the  squares  of  the  natural  num- 
bers.    Prove  these  statements. 

(4)  A  stone  dropped  into  the  vertical  shaft  of  a  mine  is  heard  to 
strike  the  bottom  after  /  seconds ;  find  the  depth  of  the  shaft,  if  the 
velocity  of  sound  be  given  =  c.    Assume  /  =  4  s.,  c—  332  metres, g  =  980. 

(5)  A  railroad  train  approaches  a  station  with  uniformly  retarded 
motion.     During  the  first  two  minutes  of  its  retarded  motion  it  makes 
3960  ft.;  during  the  next  minute  990  ft.     (a)    When  will  it  come  to 
rest?     (b)  What  is  the  retardation?     (c)  What  was  the  initial  velocity? 

(d)  When  will  its  velocity  be  4  miles  an  hour? 

(6)  Interpret  equations  (6)  and  (7)  geometrically. 

(7)  A  body  being  projected  vertically  upwards  with  an  initial  velocity 
VQ,   (a)  how  long  and  (b)  to  what  height  will  it  rise  ?     (V)  When  and 
(</)  with  what  velocity  does  it  reach  the  starting-point  ? 

(8)  A  bullet  is  shot  vertically  upwards  with  an  initial  velocity  of 
1600  ft.  per  second,     (a)  How  high  will  it  ascend?     (b)  What  is  its 
velocity  at  the  height  of  32,000  ft.?     (c)  When  will  it  reach  the  ground 
again?     (d7)  With  what  velocity?     (e)  At  what  time  is  it  32,000  ft. 
above   the  ground  ?     (/)    Explain   the   meaning   of  the  *  double  signs 
wherever  they  occur  in  the  answers. 

(9)  With  what  velocity  must  a  ball  be  thrown  vertically  upwards  to- 
reach  a  height  of  100  ft.  ? 

(TO)    A  body  is  dropped  from  a  point  A  at  a  height  AB=h  above 
the   ground ;    at    the   same   time   another  body  is   thrown   vertically 


1  1  6.]  RECTILINEAR   MOTION.  $y 

upward  from  the  point  B,  with  an  initial  velocity  v0.  (a)  When  and 
(&)  where  will  they  collide  ?  (c)  If  they  are  to  meet  at  the  height  -J-  h, 
what  must  be  the  initial  velocity  ? 

115.   The  general  problem  of  rectilinear  motion  requires  the 
integration  of  the  differential  equation 


where  j  is  a  function   of  s,  t,  and  v,  in  connection  with  the 

equation 

ds  ,  x 


As  these  two  equations  involve  four  quantities  t,  s,  v,j,  a 
third  relation  between  them,  say 

f(t,s,v,j)=o,  (9) 

is  always  necessary  in  order  to  express  three  of  these  four 
quantities  in  terms  of  the  fourth.  Next  to  the  case  of  uni- 
formly accelerated  motion  where  the  relation  (9)  is  simply 
/=  const.,  the  most  important  cases  are  those  when/  is  given 
as  a  function  of  s,  or  of  v,  or  of  both  s  and  v. 

116.  Whenever  in  nature  we  observe  a  motion  not  to  remain 
uniform,  we  try  to  account  for  the  change  in  the  character  of 
the  motion  by  imagining  a  special  cause  for  such  change.  In 
rectilinear  motion,  the  only  change  that  can  occur  in  the 
motion  is  a  change  in  the  velocity,  i.e.  an  acceleration  (or  retar- 
dation). The  cause  producing  acceleration  or  retardation  we 
call  force  (attraction,  repulsion,  pressure,  tension,  friction,  resist- 
ance of  a  medium,  elasticity,  cohesion,  etc.),  and  assume  it  to 
be  proportional  to  the  acceleration.  A  fuller  discussion  of  the 
nature  of  force  and  its  relation  to  mass  will  be  found  in  Chapter 
III.,  §  II.  The  present  remark  is  only  intended  to  make  more 
intelligible  the  physical  meaning  and  applications  of  the  prob- 
lems to  be  discussed  in  the  following  articles. 


S8 


KINEMATICS. 


[117. 


117.  Acceleration  inversely  proportional  to  the  square  of  the  dis- 
tance, i.e.  j=^i/si  where  //,  is  a  constant  (viz.  the  acceleration  at 
the  distance  s=i)  and  s  is  the  distance  of  the  moving  point 
from  a  fixed  point  in  the  line  of  motion. 

The  differential  equation  (5)  becomes  in  this  case 


(II 


the  first  integration  is  readily  performed  by  multiplying  botl 
members  by  ds/dt  so  as  to  make  the  left-hand  member  th< 
complete  derivative  of  \(ds/dt)*  or  ^v2.  Thus  we  find 


-,+<:— +c, 


where  the  constant  of  integration,  C,  must  be  determined  froi 
the  so-called  initial  conditions  of  the  problem.     For  instanc< 
if  V  =  VQ  when  s=sQt  we  have  J^02=  —fJL/sQ+C',  hence,  eliminat- 
ing C  between  this  relation  and  (i  i), 


To  perform  the  second  integration,  we  solve  this  equation  f( 
v  and  substitute  ds/dt  for  v  : 


or  putting  v£  +  2  /A/JO  =  2 


dt  /i  S 

Here  the  variables  s  and  /  can  be  separated,  and  we  find 


H9-] 


RECTILINEAR   MOTION. 


59 


To  integrate,  put  s=x?.  The  result  will  be  different  accord- 
ing to  the  signs  of  /n,  //,  and  v,  which  must  be  determined  from 
the  nature  of  the  particular  problem. 

118.  It  is  an  empirical  fact  that  the  acceleration  of  bodies 
falling  in  vacuo  on  the  earth's  surface  is  constant  only  for 
distances  from  the  surface  that  are  very  small  in  comparison, 
with  the  radius  of  the  earth.  For  larger  distances  the  acceler- 
ation is  found  inversely  proportional  to  the  square  of  the  dis- 
tance from  the  earth's  centre. 

By  a  bold  generalization  Newton  assumed  this  law  to  hold 
generally  between  any  two  particles  of  matter ;  and  this  as- 
sumption has  been  verified  by  all  subsequent  observations.  It 
can  therefore  be  regarded  as  a  general  law  of  nature  that  any 
particle  of  matter  produces  in  every  other  such  particle,  each 
particle  being  regarded  as  concentrated  at  a  point,  an  accelera- 
tion inversely  proportional  to  the  square  of  the  distance  between 
these  points.  This  is  known  as  Newton  s  law  of  universal  grav- 
itation, the  acceleration  being  regarded  as 
caused  by  a  force  of  attraction  inherent  in 
each  particle  of  matter. 

It  is  shown  in  the  theory  of  attraction 
that  the  attraction  of  a  spherical  mass, 
such  as  the  earth,  on  any  particle  outside 
the  sphere  is  the  same  as  if  the  .mass  of 
the  sphere  were  concentrated  at  its  centre. 
The  acceleration  produced  by  the  earth  on 
any  particle  outside  it  is  therefore  inversely 
proportional  to  the  square  of  the  distance 
of  the  particle  from  the  centre  of  the  earth. 


o-- 


119.    Let  us  now  apply  the  general  equa- 
tions of  Art.  117  to  the  particular  case  of  FlS-  30- 
a  body  falling  from  a  great  height  towards  the  centre  of  the 
•earth,  the  resistance  of  the  air  being  neglected. 

Let  O  be  the  centre  of  the  earth  (Fig.  30),  /\  a  point  on  its 


60  KINEMATICS.  [120. 

surface,  PQ  the  initial  position  of  the  moving  point  at  the  time 
/  =  o,  P  its  position  at  the  time  /;  let  OP^R,  OP0  =  sQ,  OP  =  s; 
and  let  g  be  the  acceleration  at  Pv  j  the  acceleration  at  P. 
Then,  according  to  Newton's  law,  j  \g=R^  :  s2.  This  relation 
determines  the  value  of  JJL  in  (10),  which  becomes 


the  minus  sign  indicating  that  the  acceleration  tends  to  dimin- 
ish the  distances  counted  from  O  as  origin. 

The  integration  can  now  be  performed  as  in  Art.  117.  Mul- 
tiplying by  ds/dt  and  integrating,  we  find  |V  =gR*/s  +  C. 
If  the  initial  velocity  be  zero,  we  have  v  =  o  for  S  =  SQ\  hence 

c=  -2>  and 


Here  again  the  minus  sign  is  selected  after  extracting  the 
square  root,  since  the  velocity  v  is  directed  in  the  sense  opposite 
to  that  of  the  distance  s. 

Substituting  ds/dt  for  v,  separating  the  variables  v  and  s,  and 
integrating,  we  find 


(.7) 


120.   Exercises. 


(1)  Find  the  velocity  with  which  the  body  arrives  at  the  surface  of 
the  earth  if  it  be  dropped  from  a  height  equal  to  the  earth's  radius,  and 
determine  the  time  of  falling  through  this  height. 

(2)  Interpret  equation  (17)  geometrically. 

(3)  Show  that  formula  (16)  reduces  to  v  =  V^p  (Art.  113)  when 
s  =  R  and  ^0  —  s  =  h  is  small  in  comparison  with  R. 


122.] 


RECTILINEAR   MOTION. 


6l 


(4)    A  particle  is  projected  vertically  upwards  from  the  earth's  surface 
with  an  initial  velocity  VQ.     How  far  will  it  rise  ? 


(5)  If,  in  (4),  the  initial  velocity  be  v0=  \gft,  how  high  and  how 
long  will  the  particle  rise  ?     How  long  will  it  take  the  particle  to  rise 
land  fall  back  to  the  earth's  surface  ? 

(6)  A  body  is  projected  vertically  upwards.     Find  the  least  initial 
^elocity    that   would    prevent  it  from    returning   to   the    earth,    taking 
r=  32,  R  =  4000  miles. 

121.  Acceleration  directly  proportional  to  the  distance,  i.e.j  =  /cs, 
where  A:  is  a  constant  and  s  is  the  distance  of  the  moving  point 
from  a  fixed  point  in  the  line  of  motion. 

The  equation  of  motion 

%-"     .'  :       (I8) 

can  be  integrated  by  the  method  used  in  Art.  117.  The  result 
of  the  second  integration  will  again  be  different  according  to 
the  sign  of  K.  We  shall  here  study  only  a  special  case,  reserv- 
ing the  general  discussion  of  this  law  of  acceleration  for  later 

(see  Arts.  177  sq.). 

122.  It  is  shown  in  the  theory  of  attraction  that  the  attrac- 
tion of  a  spherical  mass  such  as  the  earth  on  any  point  within 
the  mass  produces  an  acceleration  directed  to  the  centre  of  the 
sphere  and  proportional  to  the  distance 

from  this  centre.  Thus,  if  we  imagine 
a  particle  moving  along  a  diameter  of 
the  earth,  say  in  a  straight  narrow  tube 
passing  through  the  centre,  we  should 
have  a  case  of  the  motion  represented 
by  equation  (18). 

To  determine  the  value  of  K  for  our 
problem  we  notice  that  at  the  earth's 
surface,  that  is,  at  the  distance  OP1  =  R 
from  the  centre  O  (Fig.  31),  the  accel- 


Fig.  31. 


eration  must  be  =g.    If,  therefore,/  denote  the  numericalvalue 


62  KINEMATICS.  [122. 

of    the    acceleration    at    any    distance    OP  =  s(<R),    we   have 
j\g=s  \Rt  or  j=gs/R.     But  the  acceleration  tends  to  diminish 

the  distance  s,  hence   —  -=—  ^s.     Denoting  the  positive  cori- 
dt          R 

stant  g/R  by  ft2,  the  equation  of  motion  is 
;•'•••'     «  g=-A  where  ^  =  ^L  (19) 

Integrating  as  in  Arts.  117  and  119,  we  find 


If  the  particle  starts  from  rest  at  the  surface,  we  have  v=& 
when  s  =  R  ;  hence  o  =  —  J  p?Rz+  C  ;  and  subtracting  this  from 
the  preceding  equation,  we  find 


S*,  (20) 

where  the  minus  sign  of  the  square  root  is  selected  because 
s  and  v  have  opposite  sense. 

Writing  ds/dt  for  v  and  separating  the  variables,  we  have 


whence  /=-cos~1^s  +  Cr. 


As  s=R  when  /=o,  we  have  o=-cos~1i  +  Cr,   or 

f* 
Solving  for  s,  we  find 


(21) 
Differentiating,  we  obtain  v  in  terms  of  t\ 

(22) 


I25.] 


RECTILINEAR   MOTION. 


63. 


123.  The   motion    represented   by   equations    (21)    and    (22) 
belongs  to  the  important  class  of  simple  harmonic  motions  (see 
Arts.  177  sq.).     The  particle  reaches  the  centre  when  s  =  o,  i.e. 
when  /^=?r/2,   or   at   the   time   /=7r/2/i,.      At   this  time  the 
velocity  has  its   maximum  value.     After  passing  through  the 
centre  the  point  moves  on  to  the  other  end,  Pv  of  the  diameter, 
reaches  this  point  when  s=  —  R,  i.e.  when  yu,/=7r,  or  at  the  time 
t=TT/fj,.      As   the  velocity   then   vanishes,    the    moving   point 
begins  the  same  motion  in  the  opposite  sense. 

The  time  of  performing  one  complete  oscillation  (back  and 
forth)  is  called  the  period  of  the  simple  harmonic  motion  ;  it  is. 
evidently 

.     T=4--=-- 

2JJ,        p 

124.  Exercises. 

(1)  Equation  (19)  is  a  differential  equation  whose  general  integral 
is  known  to  be  of  the  form 

s  =  Ci  sin//,/  +  C2  cos/x/; 

determine  the  constants  Cl}  C2  and  deduce  equations  (21)  and  (22). 

(2)  Find  the  velocity  at  the  centre  and  the  period,  taking  ^=32 
and  R  =  4000  miles. 

(3)  If  the  acceleration,  instead  of  being  directed  toward  the  centre, 
is  directed  away  from  it,  the  equation  of  motion  would  ^d^s/dt^—^s  . 
Investigate  this  motion,  which  can  be  imagined  as  produced  by  a  force 
of  repulsion  emanating  from  the  centre. 


125.  Retardation  Due  to  a  Resisting  Medium.  We  know  from 
observation  that  the  velocity  of  a  body  moving  in  a  liquid  or  gas 
is  continually  diminished.  The  resistance  of  such  a  medium 
may  be  regarded  as  a  force  producing  a  retardation,  or  negative 
acceleration.  The  same  may  be  said  of  the  effect  of  friction. 
*\  The  law  according  to  which  such  resistances  retard  the  motion 
must  of  course  be  determined  by  experiment. 


«64  KINEMATICS.  [126. 

Careful  experiments  on  the  resistance  offered  by  the  air  to 
the  motion  of  projectiles  have  shown  that  this  resistance  in- 
creases with  the  quantity  of  air  displaced  ;  that  is,  with  the 
density  of  the  air,  the  cross-section  of  the  projectile,  and  the 
velocity.  The  retardation  due  to  the  resistance  of  the  air  can 
therefore  be  expressed  in  the  form 

j=>cpf(v), 

where  p  is  the  density  of  the  air,  while  K  is  a  coefficient  depend- 
ing upon  the  shape,  mass,  and  physical  condition  of  the  surface 
of  the  projectile.  Its  value  may  be  regarded  as  inversely  pro- 
portional to  the  mass  and  directly  proportional  to  the  cross- 
section  of  the  body  at  right  angles  to  the  direction  of  motion. 
The  velocity  function/^)  may  be  taken  =cv*  for  velocities 
not  exceeding  250  metres  per  second  ;  for  greater  velocities,  up 
to  about  420  metres  per  second,  it  is  proportional  to  a  higher 
power  of  v,  or  must  be  represented  by  a  more  complicated 
expression,  such  as  aiP  -\-bv-\-c\  for  velocities  above  420  metres 
it  seems  to  be  again  of  the  form  c'v2.* 

126.  Assuming  the  resistance  of  the  air  to  be  proportional  to 
the  square  of  the  velocity,  the  motion  of  a  body  falling  through 
air  of  uniform  density  is  determined  by  the  equation 


To  simplify  the  resulting  formulae,  it  will  be  convenient  to 

U? 

put  K  =  —  ,  so  that  the  equation  of  motion  is 

,     } 


dft     g 

Writing  -^  for  —  |,  the  variables  v  and  t  can  be  separated  : 
at         dp 

=dt; 


*  For  further  particulars  the  reader  is  referred  to  special  works  on  ballistics. 


127.]  RECTILINEAR  MOTION.  65 

integrating,  we  find 

,=2_log^±^,  (24) 

2fJ,          g-pV 

the  constant  of  integration  being  o  if  the  initial  velocity  be  o. 
Solving  for  z/,  we  have 

-  ~* 

.•  (25) 


As  the  numerator,  apart  from  a  constant  factor,  is  the  deriva- 
tive of  the  denominator,  the  second  integration  can  at  once  be 
performed,  giving 

J=j5  log  (**  +*-'*)  +  £ 

For  /=o,  we  have  s  =  o;  hence  o  =  ^log2  +  £7.     Hence 

t* 

*~M')-  (26) 


To  find  s  in  terms  of  v,  we  may  eliminate  dt  between  the 
fferential   equati 
resulting  equation 


-differential   equations    ds—vdt    and    dv  =  -(g*  —  i&v*)dt.      The 

o 


is  readily  integrated  ;  as  v  =  o  when  s  =  o,  we  find : 


log 

/LA  ° 

127.   Exercises. 


2  /LA2 


(1)  Show  that  as  /  increases,  the  motion  considered  in  Art.   126 
approaches  more  and  more  a  state  of  uniform  motion  without  ever 
reaching  it. 

(2)  Show  that  when  ^,  and  hence  K,  becomes  o,  the  equations  of 
Art  126  reduce  to  those  for  bodies  falling  in  vacuo. 

(3)  Investigate  the  motion  of  a  particle  thrown  vertically  upwards  in 
the  air  with  a  given  initial  velocity,  the  resistance  of  the  air  being  pro- 
portional to  the  square  of  the  velocity. 

PART  i  —  5 


66  KINEMATICS. 

(4)  Find  the  whole  time  of  ascent  in  (3)  and  the  height  to  which 
the  particle  rises. 

(5)  Show  that  owing  to  the  resistance  of  the  air  a  particle  thrown 
vertically  upwards  returns  to  the  starting  point  with  a  velocity  less  than 
the  initial  velocity  of  projection. 

(6)  A  particle  begins  moving  with  an  initial  velocity  VQ  in  a  medium 
of  constant  density  whose  resistance  is  proportional   to  the  velocity. 
Express  s  and  v  in  terms  of  /,  and  v  in  terms  of  s. 

(7)  A  body  falls  from  rest  in  a  medium  whose  resistance  is  propor- 
tional to  the  velocity.     Investigate  its  motion. 


4.  ROTATION  ;  ANGULAR  VELOCITY  ;  ANGULAR  ACCELERATION. 

128.  A  motion  of  rotation  about  a  fixed  axis  can  be  treated 
in  precisely  the  same  way  in  which  we  have  treated  rectilinear 
motion  in  the  preceding  sections.     It  is  only  to  be  remembered 
that  rotations  are  measured  by  angles  (see  Arts.  11-15),  while 
translations  are  measured  by  lengths. 

129.  The  rotation  of  a  rigid  body  (see  Art.  8)  about  a  fixed 
axis  is  said  to  be  uniform  if  the  circular  arcs  described  by  the 
same  point   in   equal   times    are    equal    throughout    the   whole 
motion;  in  other  words,  if  the  angle  of  rotation  is  proportional 
to  the  time  in  which  it  is  described.     In  this  case  of  uniform 
rotation,  the  quotient  obtained  by  dividing  the  angle  of  rotation, 
6,  by  the  corresponding  time,  /,  is  called  the  angular  velocity. 
Denoting  it  by  w  we  have  w  =  Q/t  ;  and  the  equation  of  motion  is 


Thus,  expressing  the  time  in  seconds  and  the  angle  in  radians 
(Art.  15),  the  angular  velocity  is  equal  to  the  number  of  radians 
described  per  second.  (Compare  Arts.  88-90.) 

130.  If  the  time  of  a  whole  revolution  be  denoted  by  T,  we 
have,  from  (i),  2ir=o)T'y  hence 

•'•        —  -  """   (2} 


I33-]  ROTATION.  67 

In  engineering  practice  it  is  customary  to  take  a  whole  revo- 
lution as  angular  unit  and  to  express  the  angular  velocity  of 
uniform  motion  by  the  number  of  revolutions  made  in  the  unit 
of  time.  Let  n,  N  be  the  numbers  of  revolutions  per  second 
and  per  minute,  respectively  ;  then  we  have  evidently 


(0 


131.  When  the  rotation  is  not  uniform,  the  quotient  obtained 
by  dividing  the  angle  of  rotation  by  the  time  in  which  it  is 
described,  gives  the  mean,  or  average,  angular  velocity  for  that 
time. 

The  rate  of  change  of  the  angle  of  rotation  with  the  time  at 
any  particular  moment  is  called  the  angular  velocity  at  that 
moment.  By  reasoning  in  a  similar  way,  as  in  Art.  99,  it  will 
be  seen  that  its  mathematical  expression  is 


132.   The  rate  at  which  the  angular  velocity  changes  with  the 
time  is  called  the  angular  acceleration  ;  denoting  it  by  «,  we  have 


133.  The  most  important  special  case  of  variable  angular 
velocity  is  that  of  uniformly  accelerated  (or  retarded)  rotation 
when  the  angular  acceleration  is  constant.  The  formulae  for 
this  case  have  precisely  the  same  form  as  those  given  in  Arts. 
107-1  n  for  uniformly  accelerated  rectilinear  motion.  Denoting 
the  constant  linear  acceleration  by/,  we  have,  when  the  initial 
velocity  is  o, 

FOR  TRANSLATION:  FOR  ROTATION: 

v  =jt,  co  =  at, 

e=*t\  (6) 


68  KINEMATICS.  LJ34- 

and  when  the  initial  velocities  are  VQ  and  &>0,  respectively  : 

FOR  TRANSLATION  :  FOR  ROTATION  : 


134.  Let  a  point  P,  whose  perpendicular  distance  from  the 
axis  of  rotation  is  OP=r,  rotate  about  the  axis  with  the  angular 
velocity  w  =  dB/dt.     In  the  element  of  time,  dt,  it  will  describe 
an  element  of  arc  ds=rd8=r&dt.     Its  velocity  v  =  ds/dt  (fre- 
quently  called  its  linear  velocity  in   contradistinction   to  the 
angular  velocity)  is  therefore  related  to  the  angular  velocity  of 
rotation  by  the  equation 

v=wr.  .  (8) 

135.  The  radius  vector  OP=r  sweeps  over  a  circular  sector 
which  in  uniform  rotation  has  an  area  S=^0r^  =  ^a)^,  while  in 
variable  rotation  the  infinitesimal  sector  described  during  the 
element  of  time  dt  is  dS^r^dQ^^dt. 

The  quotients 

:          '   _  |=l^  =  la,^  (9) 

for  uniform  rotation,  and 


for  variable  rotation,  represent,  therefore,  the  sectorial,  or  areal, 
velocity,  i.e.  the  rate  of  increase  of  area  with  the  time. 
The  rate  of  change  of  this  velocity  with  the  time, 


,    . 

dt*dt      ' 


is  called  the  sectorial,  or  areal,  acceleration. 


136.]  ROTATION.  69 

136.   Exercises. 

(1)  If  a   fly-wheel   of   12    ft.    diameter   makes   30   revolutions   per 
minute,  what  is  its  angular  velocity,  and  what  is  the  linear  velocity  of  a 
point  on  its  rim? 

(2)  A  pulley  5  ft.  in  diameter  is  driven  by  a  belt  travelling  500  ft. 
a  minute.     Neglecting  the  slipping  of  the  belt,  find   (a)   the  angular 
velocity  of  the  pulley  in  radians,  and  (^)  its  number  of  revolutions  per 
minute. 

(3)  Find  the  constant  acceleration  (such  as  the  retardation  caused  by 
a  Prony  brake)  that  would  bring  the  fly-wheel  in  Ex.  (i)  to  rest  in  1 
minute. 

(4)  How   many  revolutions   does   the   fly-wheel  in   Ex.   (3)   make 
during  its  retarded  motion  before  it  comes  to  rest? 

(5)  A  wheel   is  running  at  a  uniform  speed  of  32  turns  a  second 
when  a  resistance  begins  to  retard  its  motion  uniformly  at  the  rate  of  8 
radians  per  second,     (a)  How  many  turns  will  it  make  before  stopping? 
(l>)  In  what  time  is  it  brought  to  rest  ? 

(6)  A  belt  runs  over  two  pulleys  turning  about  parallel  axes.     Show 
that  the  angular  velocities  of  the  pulleys  are  inversely  proportional  to 
their  diameters.     Do  the  pulleys  rotate  in  the  same  or  opposite  sense  ? 


70  KINEMATICS.  [137. 


III.   Plane  Kinematics. 

I.    VELOCITY  J   COMPOSITION    OF    VELOCITIES  ;    RELATIVE 
VELOCITY. 

137.  The  motion  of  a  point  in  a  curved  path  would  not  be 
completely  characterized   by  its   velocity  and   acceleration    as 
defined  in  the  preceding  section  ;  the  varying  direction  of  the 
motion,  and  the  rate  of  change  of  direction,  must  be  taken  into 
account.     It  is   convenient  to  incorporate  these  ideas   in  the 
definitions  of  velocity  and  acceleration.     By  this  generalization 
of   their   original   meaning,  velocity  and   acceleration   become 
vectors,  i.e.  magnitudes  having  both  length  and  direction. 

138.  The  generalized  idea  of  velocity  as  a  vector  may  be 
derived  as  follows  : 

Consider  a  point  P  moving  in  a  curve  (Fig.  32).     Let  P  be 

its  position  at  the  time  t,  P' 
its  position  at  the  time  /+  A/, 
and  let  PQP  =  s)  PP'  =  bs. 
The  space  s  described  in  any 
time  /  may  be  regarded  as 
some  function  of  the  time  /, 
say  *=/(/). 

The  mean  velocity 


Fi     32>  for  the  time  A/  during  which 

the  point  passes  from  P  to 

P'may  be  represented  by  a  length  PS  laid  off  on  the  chord 
PP'  from  P.  As  A/  diminishes,  P'  approaches  P,  and  in  the 
limit  when  Aj/A/  becomes  the  derived  function  ds/dt=f'(t), 
the  chord  merges  into  the  tangent  at  P.  This  leads  us  to  rep- 
resent the  velocity  at  the  time  t,  or  at  the  place  P,  by  a  length 
PT  proportional  to  ds/dt  laid  off  on  the  tangent  at  P  from  this 
point  in  the  sense  of  the  motion.  The  vector  PT  will  then 
completely  represent  the  velocity  at  the  time  /. 


I4I-] 


VELOCITY. 


139.  The  vector  PTma.y  also  be  regarded  as  the  limit  of  a  vector 
PS  laid  off  on  the  chord  PP  as  before,  but  proportional  to  the  velocity 
with  which  the  point  would  describe  the  chord  PP  in  the  time  A/,  i.e. 

to  the  velocity  PS=— For  as  A/  approaches  the  limit  o, 

A/ 

PS  approaches  the   direction  of  the  tangent,  and  the  ratio  of  the  arc 
AJ  to  the  chord  PP  approaches  the  limit   i.      Hence  the  equation 

—  =  — — - — -  •  PS  gives  in  the  limit  lim  —  =  lim  PSt  or  PT—  lim  PS. 
A/      chord  PP  A/ 

It  may  be  noticed  here  that,  in  view  of  the  practical  applications,  the 
function /(/)  =  s  is  in  mechanics  always  supposed  to  be  itself  continuous 
and  to  possess  continuous  and  finite  derivatives  of  the  first  and  second 
order. 

140.  Velocity  having  thus  been  denned  as  a  vector,  we  may 
at  once  apply  to  it  the  rules  for  vector  composition  and  vector 
resolution  laid  down  in  Arts.  45-55  for  vectors  representing  dis- 
placements.     Thus    if   a  point   be  subjected  to  two  or  more 
simultaneous  velocities,  the  velocity  of  the  resulting  motion  will 
be  represented  by  the  vector  found  by  geometrically  adding  the 
component  velocities.      A  velocity  may  be  resolved  into  any 
number  of  component  velocities  whose  geometrical  sum  is  equal 
to  the  given  velocity. 

141.  We  proceed  to  consider  the  most  important  cases  of 
resolution  of  a  velocity  in  a 

plane. 

Let  a  point  P  move  in  a 
curve  PQP  (Fig.  33)  whose 
equation  is  referred  to  rec- 
tangular Cartesian  co-ordi- 
nates xy  y.  It  is  usually  con- 
venient in  this  case  to 
resolve  the  velocity  v  par-  ' 


into    vx 


Fig.  33. 


allel    to    the    axes 
and  vy. 

If  a  be  the  angle  made  by  the  vector  v  with  the  axis  of  x,  we 
have  vx  =  vcQsa,  v,  =  vsina.  And  as  the  element  ds  of  the 
curve  at  P  makes  the  same  angle  a  with  the  axis  of  x,  we  also 


72  KINEMATICS.  [142. 


have  dx=ds  cos  a,  dy=ds  sin  a.     Divid|    I  by  dt  and  comparing 
with  the  preceding  equations,  we  find 


Conversely,  knowing  the  velocities  of  the  moving  point  paral- 
lel to  the  axes,  we  find  its  resulting  velocity  from  the  relation 


7i=J(^V#Y 
•   ^\dti+\dt)- 


(2) 


142.  If  the  equation  of  the  path  be  given  in  polar  co-ordinates, 
it  may  be  convenient  to  resolve  the  velocity  v  along  the  radius 
vector  OP  and  at  right  angles  to  it  (Fig.  34). 


Fig.  34. 

Let  r,  6  be  the  polar  co-ordinates,  a  the  angle  between  v  and  r', 
then  vr=v  cos  a,ve  =  v  sin  a.  The  element  ds  of  the  curve  has  in 
the  same  directions  the  components  dr=dscosa,  rd0  = 
Hence,  dividing  by  dt,  we  find 


and 


v  = 


143.  In  the  case  of  relative  motion  we  have  to  distinguish 
between  the  absolute  velocity  v  of  a  point,  its  relative  velocity  vl9 
and  the  velocity  of  the  body  of  reference  v2. 


I44-]  VELOCITY.  73 

To  fix  the  ideas,  imagine  a  man  walking  on  deck  of  a  steam- 
boat. His  velocity  of  walking  is  his  relative  velocity  z/j;  the 
velocity  of  the  boat  (say  with  respect  to  the  water  or  shore 
regarded  as  fixed),  or  more  exactly  speaking,  the  velocity  of  that 
point  of  the  boat  at  which  the  man  happens  to  be  at  the  time, 
is  the  velocity  z/2  of  the  body  of  reference  ;  and  the  velocity  with 
which  the  man  is  moving  with  respect  to  the  water  or  shore,  is 
his  absolute  velocity. 

Representing  these  three  velocities  by  means  of  their  vectors, 
we  evidently  find  the  absolute  velocity  v  as  the  geometric  sum  of 
the  relative  velocity  Vj  and  the  velocity  v2  of  the  body  of  reference, 
just  as  in  the  case  of  displacements  of  translation  (Art.  53). 
And  conversely,  the  relative  velocity  is  found  by  geometrically 
subtracting  from  the  absolute  velocity  the  velocity  of  the  body  of 
reference. 

It  is  often  convenient  to  state  the  last  proposition  in  a  some- 
what different  form.  Imagine  that  we  give  the  velocity  —  z/2. 
both  to  the  man  and  to  the  boat ;  then  the  boat  is  brought  to 
rest,  and  the  resulting  velocity  of  the  man  is  what  was  before 
his  relative  velocity.  Hence  the  relative  velocity  is  found  as  the 
resultant  of  the  absolute  velocity,  and  the  velocity  of  the  body  of 
reference  reversed. 

144.   Exercises. 

(1)  A  straight  line  in  a  plane  turns  with  constant  angular  velocity  o> 
about  one  of  its  points  O,  while  a  point  P,  starting  from  O,  moves  along 
the  line  with  a  constant  velocity  VQ.     Determine  the  absolute  path  of 
P  and  its  absolute  velocity  v. 

(2)  Show  how  to  construct  the  tangent  and  normal  to  the  spiral  of 
Archimedes  r  =  aO,  where  9  =  o>/. 

(3)  A  wheel  of  radius  a  rolls  on  a  straight  track  with  constant  velocity 
(of  its  centre)  z/0.     Find  the  velocity  v  of  a  point  /'on  the  rim. 

(4)  Show  that  the  tangent  to  the  cycloid  described  by  P,  Ex.  (3), 
passes  through  the  highest  point  of  the  wheel. 

(5)  Show  that  the  tangent  to  the  ellipse  bisects  the  angle  between 
the  radii  vectores  r,  r<  drawn  from  any  point  P  on  the  ellipse  to  the 
foci  S,  8. 


74  KINEMATICS.  [145. 

(6)  Construct  the  tangent  to  any  conic  section  when  a  directrix  anc 
the  corresponding  focus  are  given. 

(7)  Two  trains  of  equal  length  pass  each  other  with  equal  velocity 
-on  parallel  tracks.     A  man  riding  on  a  bicycle  along  the  track  at  the 
rate  of  8  miles  an  hour  notices  that  the  train   meeting  him  takes 
seconds   to   pass   him,  while    the   other  takes    6    seconds.     Find  the 
velocity  of  the  trains. 

(8)  A  swimmer,  starting  from  a  point  A  on  one  bank  of  a  river 
wishes  to  reach  a  certain  point  B  on  the  opposite  bank.     The  velocity 
#2  of  the  current  and  the  angle  0  made  by  AB  with  the  direction  of  the 
-current  being  given,   determine  the  least   relative   velocity  i\  of  the 
swimmer  in  magnitude  and  direction. 

(9)  Two  men,  A  and  B,  walking  at  the  rate  of  3  and  4  miles  an  hour 
respectively,  cross  each  other  at  a  rectangular  street  corner.     Find  the 
relative  velocity  of  A  with  respect  to  B  in  magnitude  and  direction. 

(10)  A  man  jumps  from  a  car  at  an  angle  of  60°  with  a  velocity  o 
&  feet  a  second  (relatively  to  the  car).     If  the  car  be  running  10  miles 
an  hour,  with  what  velocity  and  in  what  direction  does  the  man  strike 
the  ground? 

(n)  The  point  J\  moves  with  constant  velocity  v±  along  the  line 
PiQ.  In  what  direction  JP2Q  must  a  point  P2  move  with  constan 
velocity  v.2  in  order  to  meet  PJ  What  is  the  locus  of  Q  when  the 
direction  of  Pl  Q  varies  ?  When  is  the  solution  impossible  ? 

(12)  A  point  Amoves  uniformly  in  a  circle,  while  another  point 
moves  with  equal  velocity  along  a  tangent  to  the  circle.     Find  the 
relative  path  of  either  point  with  respect  to  the  other. 

(13)  The  velocity  of  light  being  taken  as  300,000  kilometres  per  sec 
ond,  and  the  velocity  of  the  earth  in  its  orbit  as  30  kilometres,  determine 
approximately  the  constant  of  the  annual  aberration  of  the  fixed  stars. 

2.     APPLICATIONS. 

145.  The  motion  of  the  piston  of  a  steam  engine  furnishes 
interesting  illustrations  of  the  application  of  graphical  methods 
in  kinematics. 

In  Fig.  35,  let  OQ=a  be  the  crank  arm,  PQ  =  l=ma  thi 
-connecting  rod,  P1P2  =  s  the  " stroke,"  so  that  l=ma  =  %m 


H7-] 


PLANE   MOTION. 


75 


As  P1P2  =  A1A2  =  2a,  we  may  regard  A^A^  as  representing  the 
stroke.  The  position  of  the  piston  head  P  at  the  time  when  the 
crank  pin  is  at  Q  will  then  be  found  as  the  intersection  N  oi  a 
circle  of  radius  /  described  about  P  with  the  diameter  A 


Fig.  35. 

of  the  crank  circle  ;  in  other  words,  N  represents  the  position  of 
the  piston  corresponding  to  the  angle  A1QQ  =  0  in  the  forward 
stroke  and  to  the  angle  A1OQf  =  27r  —  0  in  the  'return  stroke. 

146.  The  crank  may  generally  be  assumed  to  turn  uniformly, 
making  n  revolutions  per  second.  The  linear  velocity  of  the 
crank  pin  Q  is  therefore  u  =  2Tra  •  n  =  Trns. 

For  the  piston  head  Pt  or  for  the  point  N,  we  must  distin- 
guish between  its  mean,  or  average,  velocity  V,  and  its  variable 
instantaneous  velocity  v  at  any  particular  moment.  For  each 
revolution  of  the  crank  the.  piston  head  completes  a  double  stroke 
so  that  its  mean  speed  is  V—2ns.  Hence  we  have 


u 


_  TT 
2ns~  2 


147.  The  instantaneous  velocity  v  of  the  piston  can  be  found 
graphically  by  considering  the  motion  of  the  connecting  rod 
PQ.  The  velocity  u  of  the  end  Q  is  known,  both  in  magnitude 
and  direction  ;  the  velocity  v  of  the  other  end  is  known  in  direc- 
tion only.  Now  considering  that  the  length  of  the  rod  PQ  is 
invariable  and  hence  the  components  of  u  and  v  along  PQ  must 


76 


KINEMATICS. 


[148. 


be  equal,  we  can  find  the  magnitude  of  v  by  drawing  (Fig.  36} 
from  any  point  M  parallels  to  u  and  v,  laying  off  u  to  scale  and 
drawing  through  its  extremity  a  perpendicular  to  the  direction  of 
PQ ;  this  perpendicular  will  cut  off  the  proper  length  on  the 
direction  of  v. 


Fig.  36. 

In  applying  this  construction  to  our  case  it  will  be  convenien 
to  turn  the  auxiliary  diagram  of  velocities  by  an  angle  of  90 
and  place  it  so  as  to  make  M  coincide  with  O  ;  u  will  then  lie| 
along  OQ,  and  v  at  right  angles  to  OP.     Hence,  if  the  scale  ol 
velocities  be  selected  so  as  to  have  u  represented  in  length  by 
OQ,  v  will  be  represented  on  the  same  scale  by  OR,  that  is,  b4 
the  segment  cut  off  by  PQ  produced  on  the  perpendicular  to- 
OP  drawn  through  O. 


148.  The  variation  of  the  piston  velocity  in  the  course  of 
motion  can  best  be  exhibited  graphically.  Thus  a  polar  curved 
of  piston  velocity  is  obtained  by  laying  off  on  OQ  a  length 
OR'  =  OR,  for  a  number  of  positions  of  OQ,  and  joining  the 
points  R'  by  a  continuous  curve. 

Another  convenient  method  consists  in  erecting  perpendicu- 
lars to  OP  at  the  various  positions  of  P  and  laying  off,'  on  these 
perpendiculars,  OR"  =  OR  —  v. 


149.  To  derive  an  analytical  expression  for  the  piston  velocit] 
v,  let  $  be  the  angle  OPQ  which  determines  the  position  of  thi 
connecting  rod. 

It  follows  from  the  construction  of  the  velocity  v  given  \\ 
Art.  147  (see  Fig.  36)  that 


152.]  PLANE    MOTION. 

v     OR     sin  (d  +  6) 
'   =        =  - 


If,  as  is  usually  the  case,  the  connecting  rod  is  much  longer 
than  the  crank  arm,  </>  will  be  a  small  angle,  and  we  may  substi- 
tute sin  (f>  for  tan  0.  But  from  the  triangle  OPQ  we  have 


Hence  v  =  u( sin0  +  cos0-  —  sin0)  =  u(  sin  0  +  —  sin  20V 
\  m        J       \  2m  J 

150.  The  motion  of  the  piston  head  being  rectilinear,  we  find 
its  acceleration  j  by  differentiating  the  expression  for  v  found 
in  the  preceding  article  with  respect  to  t : 

.    dv     f  .    n       I  *\du        f       .      i  A 

;  =  -T  =  sm0H sm20)-7-  +  «|  cos0H — cos  20 

*     dt    \  2m  )  dt        \  m  J 


J0 
dt 


or,  since    —  =  &>  = 


/=(  sin  0H sin  2  0 ) -¥  +  ( cos  0H —  cos  2  0  ]— > 

\  2m  J  dt     \  m  )  a 

where  — =o  if  the  crank  motion  can  be  regarded  as  uniform. 

151.  If  the  connecting  rod  were  of  infinite  length  so  as  to 
make  PQ  (in  Fig.    35)    parallel  to  A^A^   the  position  of  the 
piston  corresponding  to  the  position  Q  of  the  crank  pin  would 
l>e  represented  by  the  projection  M  of  Q  on  A^A^  ;  that  is,  NM 
would  be  =  o.     This  length  NM  is  therefore   called  the  devia- 
tion due  to  the  obliquity  of  the  connecting  rod. 

With  NM=o  the  expression  for  the  acceleration  (Art.  150) 
would  reduce  to  dv/dt—(t^la)  cos0,  representing  a  simple  har- 
monic motion  (see  Art.  179). 

152.  The  slide  valve  of  a  steam  engine  is  generally  worked 
by  an  eccentric  whose  radius  is  set  on  the  shaft  at  such  an 


KINEMATICS. 


angle  as  to  shut  off  the  steam  when  the  crank  makes  a  certain 
angle  0  with  the  direction  of  motion  of  the  piston.  It  fol- 
lows that  the  fraction  of  stroke  completed  before  cut-off  takes 
place  is  affected  by  the  obliquity  of  the  connecting  rod.  The 
rates  of  cut-off  are  therefore  different  in  the  forward  and  back- 
ward strokes.  In  the  forward  stroke,  the  effect  of  the  obliquity 
is  to  put  the  piston  in  advance  of  the  position  it  would  have 
if  the  connecting  rod  were  of  infinite  length  ;  in  the  return, 
stroke,  i.e.  when  6  is  greater  than  TT,  the  piston  lags  behind. 

153.    An  analytical  expression  for  the  deviation  due  to  obliquil 
is  readily  obtained  from  Fig.  35.     We  have 

MN=  PN-  PM=  I  ( i  -  cos 

.   20     msf     .    < 

=  ms  sm^  -  =  —  (2  sm  —    > 
2       4\ 

or  approximately,  since  <f>  is  small, 


ms  . 
'  4  S11 


Also,  as  in  Art.  149,  — -£=— ; 
sin  u     m 


hence 


s 
4m 


The  greatest  value  of  J/7Vis  thus  seen  to  be  s/^m ;  for  instanct 
if  the  connecting  rod  be  four  times  the  length  of  the  crank,  th< 
deviation  due  to  obliquity  cannot  exceed  1/16  of  the  stroke. 

154.   Exercises.* 

1 i )  Construct  a  polar  diagram  exhibiting  the  position  of  the  piste 
for  all  angles  0  by  laying  off  on  the  crank  arm  O  Q  a  length  ON]  = 
and  joining  the  points  N1  by  a  continuous  curve. 

(2)  Construct  the  curves  of  piston  velocity  indicated  in  Art.  148. 

*  These  problems  are  taken  with  slight  modification  from  COTTERILL'S  Applii 
mechanics,  1884,  p.  112. 


155-3 


ACCELERATION. 


(3)  Show  that  for  a  connecting  rod  of  infinite  length  the  two  loops 
of  the  curve  of  Ex.  i  reduce  to  two  equal  circles. 

(4)  The  driving  wheels  of  a  locomotive  are  6  ft.  in  diameter;  find 
the  number  of  revolutions  per  minute  and  the  angular  velocity,  when 
running  at  50  miles  per  hour.     If  the  stroke  be  2  ft.,  find  the  speed  of 
the  piston. 

(5)  The  pitch  of  a  screw  is  24  ft.,  and  the  number  of  revolutions  70 
per  minute.     Find  the  speed  in  knots.     If  the  stroke  is  4  ft.,  find  the 
speed  of  piston  in  feet  per  minute. 

(6)  The  stroke  of  a  piston  is  4  ft.,  and  the  connecting  rod  is  9  ft. 
long.     Find  the  position  of  the  crank,  when  the  piston  has  completed 
the  first  quarter  of  the  forward  and  backward  strokes  respectively.    Also 
find  the  position  of  the  piston  when  the  crank  is  upright. 

(7)  The  valve  gear  is  so  arranged  in  the  last  question  as  to  cut  off 
the  steam  when  the  crank  is  45°  from  the  dead-points  both  in  the  for- 
ward and  backward  strokes.     Find  the  point  at  which  steam  will  be  cut 
off  in  the  two  strokes.     Also  when  the  obliquity  of  the  connecting  rod 
is  neglected. 

3-     ACCELERATION    IN    CURVILINEAR    MOTION. 

155.  Let  the  velocity  of  a  moving  point  be  represented  by 
the  vector  v  =  PTat  the  time  /, 
and  by  the  vector  vt  =  P'T1  at 
the  time  /+  A^  (Fig.  37).  Then, 
drawing  from  any  point  O  OV 
and  OV  respectively  equal  and 
parallel  to  FT  and  P'Tf,  the 
vector  W  represents  the  geo- 
metrical difference  between  vr 
and  v  ;  in  other  words,  VV1  is 
the  velocity  which,  geometrically 
added  to  v,  produces  v'.  The 
vector  VV  approaches  the  limit 
o  at  the  same  time  with  A/  and 


Fig.  37. 


PP'.     This  limit  of  VV  for  an  infinitely  small  time  dt  may  be 
called   the  geometrical  differential  or  vector  differential,  of  v. 


So 


KINEMATICS. 


[156. 


Dividing  this  infinitesimal  vector  by  dt,  we  obtain  in  general  a 

finite   magnitude    — ,  the  geometrical  derivative   of    the 

dt 

velocity  with  respect  to  the  time,  and  that  is  what  we  call  the 
acceleration  at  the  time  t  or  at  the  point  P.  We  represent  it 
geometrically  by  a  vector  j  drawn  frotn  P  parallel  to  the  direc- 
tion of  Km  VV. 

It  will  be  noticed  that  the  sense  of  the  acceleration  will  be 
towards  that  side  of  the  tangent  of  the  curve  on  which  the 
centre  of  curvature  is  situated. 

156.  Suppose  a  point  P  to  move  along  a  curve  P^P^P^  ...i 
with  variable  velocity  v  (Fig.  38).  From  any  fixed  origin  O\ 
draw  a  vector  OVl  =  vlJ  equal  and  parallel  to  the  velocity  v^  ofj 


Plt  and  repeat  this  construction  for  every  position  of  the  mov- 
ing point  P.  The  ends  P\,  V^  V&  ...  of  all  these  radii  vectores 
drawn  from  O  will  form  a  continuous  curve  V^V^Y^...  which  is 
called  the  hodograph  of  the  motion  of  P. 

•  If  we  imagine  a  point  V  describing  this  curve  V^V^V^...  at 
the  same  time  that  P  describes  the  curve  P^PJP^  .  ..,  it  is  evident 


that  the  velocity  of  Vt  i.e.  fi 


dt 


!,  laid  off  on  the  tangent  of  I 


the  curve 


...,  represents  the  acceleration  of  the  point 


I59-] 


ACCELERATION. 


8l 


both  in  magnitude  and  direction  ;  i.e.  the  velocity  of  the  hodo- 
graph  is  the  acceleration  of  the  original  motion. 

It  is  easy  to  see  how  these  considerations  might  be  extended. 
We  might  construct  the  hodograph  of  the  hodograph ;  its 
velocity  might  be  called^  the  acceleration  of  the  second  order  for 
the  motion  of  P  ;  and  so  on. 

It  is  sometimes  convenient  to  draw  the  radii  vectores  of  the 
hodograph  not  parallel  to  the  velocities  of  the  point  P,  but  so  as 
to  make  some  constant  angle  (usually  a  right  angle)  with  these 
velocities. 

157.  Exercises. 

(1)  Discuss  rectilinear  motion  as  a  special  case  of  plane  motion. 

(2)  Show  that  the  hodograph  of  rectilinear  motion  is  a  straight  line. 

(3)  Show  that  the  velocity  of  a  moving  point  is  increasing,  constant, 
or  diminishing,  according  to  the  value  of  the  angle  if/  between  v  and  j 
(Fig.  37)- 

158.  Acceleration  having  been  defined  as  a  vector,  the  rules 
for  vector  composition  and  resolution  may  be  applied  to  accelera- 
tion just  as  they  were  before  applied  to  displacements  and  to 
velocities.    Thus,  a  point  subjected  to  two  or  more  simultaneous 
accelerations  will  have  a  resulting  acceleration  found  by  geo- 
metrically adding  the  component  accelerations  ;  and  conversely, 
the  acceleration  of  a  point  may  be  resolved  in  various  ways. 

159.-  Let  the  vector  /  which  represents  the  acceleration  of 
the  point  P  at  the  time  /,  make  an  angle  -^  with  the  vector 
representing  the  velocity  v  at  the  same  time  (see  Fig.  37). 
Resolving  the  vector  j  along  the  tangent  and  normal  at  P,  we 
obtain  the  tangential  acceleration  jt=jcos-^  and  the  normal 
acceleration  /„=/  sin  i/r. 

To  find  expressions  for  these  components,  let  us  regard  PP? 
in  Fig.  37  as  the  element  ds  of  the  path  described  by  P ;  then 
the  length  of  P'T',  or  of  OV\  is  v'  =  v  +  dv,  and  the  angle 
VOV1,  being  equal  to  the  angle  between  two  consecutive 

PART  / — 6 


82 


KINEMATICS. 


[i  60. 


tangents  of  the  curve,  is  the  angle  of  contingence  da.  at  P. 
This  angle  being  equal  to  the  angle  between  the  normals  at  P 
and  P',  we  have  pda  =  ds,  where  p  is  the  radius  of  curvature 
at  P. 

Resolving  the  elementary  acceleration,  i.e.  the  infinitesimal 
vector  FF',  along  <9Fand  at  right  angles  to  OF,  we  find  the 
components  FF'  cos^  =  ^,  FF'  sin  ^r  =  vda  —  vds/p.  Dividing 
by  dt  and  observing  that  ds/dt=v,  we  finally  obtain 


dv 
dt 

da 


By  composition  we  have 


(2) 


(3) 


160.  When  rectangular  Cartesian  co-ordinates  are  used,  we 
may  resolve  the  acceleration  j  into  two  components  jx=j  cos  <j>, 
j9=jsm<l>  parallel  to  the  co-ordinate  axes  Ox,  Oy\  <f>  being  the 
angle  made  by  the  vector  j  with  the  axis  of  x.  We  obtain  an 
expression  for  jx  by  projecting  the  infinitesimal  triangle  OVVf 
(Fig.  37)  on  the  axis  Ox  and  denoting,  as  before,  the  projections 
of  the  velocities  OV,  OV  by  vx,  v'x.  This  gives 


VV  cos  ^  —  v!x  —  vx  =  dv# 

^, 

whence,  dividing  by  dt,  jx  =  dvjdt.    Similarly,  we  fi  nd  j\ 
Hence,  by  formulae  (i),  Art  141, 


dt       dt* 


Jy     ~df      dt* 


dvjdt. 


(4) 


These  so-called  equations  of  motion  offer  the  advantage  that 
the  curvilinear  motion  is  replaced  by  two  rectilinear  motions, 
thus  avoiding  the  use  of  vectors. 


i6i .]  ACCELERATION. 

By  composition,  we  have  of  course 


7  = 


(5) 


161.  For  polar  co-ordinates  r,  0,  we  may  resolve  the  accelera- 
tion j  into  a  component  jv  along  the  radius  vector  r  and 
a  component  je  at  right 

angles   to   r.     Expressions  3y 

for  these  components  are 
readily  found  by  projecting 
the  components 


on  r  and  at  right  angles  to 
r  (Fig.  39)  • 


Fig.  39. 


Differentiating  the  relations  x=rcos6,  y  =  rsin6,  we  find 


dx     dr        /! 
—  =  —  cos#  — 
dt     dt 


and  differentiating  again  : 


dQ     dv     dr 

—  ,    -^-  =  — 

dt     dt     dt 


* 

r  cos  6—-  ; 
dt 


dt* 


dt  dt 


Substituting  these  expressions  for  —  ^  and 
equations  for  jnjB,  we  find  : 


in   the   above 


d 


,~ 


84  KINEMATICS.  [162. 

162.  The  meaning  of  these  expressions  will  perhaps  be  better 
understood  by  the  following  geometrical  derivation.  As  shown 
in  Art.  142,  the  velocity  v  has  the  components 

dr  <te 


the  former  along  the  radius  vector,  the  latter  at  right  angles 
to  it.  During  the  element  of  time  dt,  while  the  moving  point 
passes  from  P  to  P'  (Fig.  40),  each  of  the  vectors  vr,  VQ 


Fig.  40. 

receives  a  geometrical  increment  VrV'n  VeVl  V  Let  us  resolve 
each  of  these  infinitesimal  vectors  along  r  and  at  right  angles 
to  r,  and  the'n  combine  the  two  components  along  r,  and  also 
the  two  components  perpendicular  to  r\  finally,  dividing  by  dt, 
we  obtain/.  and/fl. 

Thus  vr  gives  —  £  along  ry  and  —  —  at   right   angles    to    ry 
dt*  dt  dt 


hile  z/0,  or  r-—,  contributes  —  r  I  —  )  along  r  and 
dt  \dt  ) 


w 


^^—^  =  ——4- 
dt\  dt)    dt  dt       dt* 

at  right  angles  to  r.     We  obtain  in  this  way  the  same  expres- 
sions for/,.,/^  as  in  the  formulae  (6)  above. 


i64.] 


PLANE   MOTION. 


163.  Exercises. 

(1)  Show  that  the  sectorial  velocity  (Art.  135)  is  constant  whenever 

J9  =  °" 

(2)  Show  that  the  normal  component  of  the  acceleration   is  the 
product  of  the   radius  of  curvature   into  the   square   of  the  angular 
velocity  about  the  centre  of  curvature. 

(3)  Show  that  the  velocity  is  the   mean  proportional  between  the 
acceleration  and  half  the  chord  intercepted   by  the  direction  of  the 
acceleration  on  the  osculating  circle. 

(4)  If  the  acceleration  of  a  point  P  be  always  directed  to  a  fixed 
point  Ay  show  that  the  radius  vector  ^/*  describes  equal  areas  in  equal 
times. 

(5)  Show  that  in  uniform  circular  motion  the  acceleration  is  directed 
to  the  centre  and  proportional  to  the  radius. 

(6)  A  wheel  rolls  on  a  straight  track;  find  the  acceleration  of  its 
lowest  point. 

4.     APPLICATIONS. 

164.  Inclined  Plane.     Imagine  a  body  sliding  down  a  smooth 
plane  inclined  at  an  angle  6  to  the  horizon.    In  addition  to  the 
assumptions  made  in  the  case  of  falling  bodies  (see  Art.  112) 
we  assume  that  the  motion  takes  place  along  a  "  line  of  greatest 
slope,"  i.e.  in  a  vertical  plane  at  right  angles  to  the  intersection 
of  the  inclined  plane  with  a  horizontal   plane.     A  "  smooth  " 
plane  means  one  that  offers  no  fric- 

tional  resistance.  The  body  is  there- 
fore subject  only  to  the  acceleration 
of  gravity,  g\  and  it  is  sufficient  to 
consider  the  motion  of  a  single  point 
of  the  body. 

Resolving  g  into  two  components, 
gcosO  perpendicular  to  the  plane 
and  £-'sin0  along 'the  plane  (Fig.  41), 


Fig.  41, 


it  will  be  seen  that  the  former  component,  being  at  right  angles 
to  the  velocity,  cannot  change  the  magnitude  of  this  velocity. 


86  KINEMATICS.  [165. 

We  have  therefore  simply  a  rectilinear  motion  with  the  constant 
acceleration  gs'mO,  so  that  all  the  formulae  of  Art.  107-113 
will  here  apply  if  for  the  acceleration  j  (or  g)  we  substitute 
£"sin#. 

Thus,  if  the  initial  velocity  be  o,  the  motion  is  determined 
by  the  equations 

(i) 

*,  (2) 

.  (3) 

165.    If  there  be  an  initial  velocity  T/O  parallel  to  the  line  of 
greatest  slope  of  the  inclined  plane,  the  equations  are 


(I') 

«,  (2') 


where  ^0  is  to  be  regarded  as  positive  if  its  direction  is  down 
the  plane  and  negative  when  up  the  plane.  • 

If  the  initial  velocity  VQ  be  inclined  to  the  plane  at  an 
angle  ft,  it  can  be  resolved  into  the  components  z>0cos/3  and 
VQ  sin  ft,  the  former  alone  being  effective  so  that  v§  cos  ft  must 
be  substituted  for  v§  in  the  above  formulae. 

166.   Exercises. 

(1)  A  railroad  train  is  running  up  a  grade  of  i  in  250  at  the  rate  of 
15  miles  an  hour  when  the  coupling  of  the  last  car  breaks.     Neglecting 
friction,  (a)  how  far  will  the  car  be  after  two  minutes  from  the  point 
where  the  break  occurred  ?     (<£)  When  will  it  begin  moving  down  the 
grade?     (c)   How  far  behind   the  train  will   it   be   at   that   moment? 
(d)   If  the   grade   extend    2000  ft.  below  the  point  where  the  break 
occurred,  with  what  velocity  will  it  arrive  at  the  foot  of  the  grade  ? 

(2)  Show  that  the  final  velocity  is  independent  of  the  inclination  of 
the  plane;  in  other  words,  in  sliding  down  a  smooth  inclined  plane  a 


167.]  PLANE   MOTION.  87 

body  acquires   the  same  velocity  as    in    falling  vertically  through  the 
•"height"  of  the  plane. 

(3)  Show  that  it  takes  a  body  twice  as  long  to  slide  down  a  plane 
•of  30°  inclination  as  it  would  take  it  to  fall  through  the  height  of  the 
plane. 

(4)  At  what  angle  6  should  the  rafters  of  a  roof  of  given  span  2  b  be 
inclined  to  make  the  water  run  off  in  the  shortest  time  ? 

(5)  Prove  that  the  times  of  descending  from  rest  down  the  chords 
issuing  from  the  highest  (or  lowest)  point  of  a  vertical  circle  are  equal. 

(6)  If  any  number  of  points  starting  at  the  same  time  from  the  same 
point  slide  down  different  inclined  planes,  they  will  at  any  time  /  all  be 
situated  on  a  sphere  passing  through  the  starting  point  and  having  a 
diameter  = 


(7)  Show  how  to  construct 'geometrically  the  line  of  quickest  descent 
from  a  given  point :  (a)  to  a  given  straight  line,  (b)  to  a  given  circle, 
situated  in  the  same  vertical-;plaiie. 


(8)  Analytically,  the  line  &r  quickest  or  slowest  descent  from  a  given 
point  to  a  curve  in  the  sanje  vertical   plane  is    found  by   taking  the 
•equation  of  the  curve  in  polar  co-ordinates,  r=/(0),  with  the  given 
point  as  origin  and  the  axis  horizontal.     The  time  of  descending  the 
radius  vector  r  is  't  =  ~^~2r/(g  sin  6).    Show  that  this  becomes  a  maxi- 
mum or  minimum  when  tan  0=/(0)//'(0),  according  as /(0) +/"(#) 
is  negative  or  positive. 

(9)  Show  that  the  line  of  quickest  descent  to  a  parabola  from  its 
focus,  the  axis  of  the  parabola  being  horizontal,  is  inclined  at  an  angle 
of  60°  to  the  horizon. 


167.  Projectiles.  With  the  same  assumptions  as  in  Art.  112, 
the  motion  of  a  projectile  reduces  to  that  of  a  point  subject  to 
the  constant  acceleration  of  gravity  and  starting  with  an  initial 
velocity  VQ  inclined  to  the  horizon  at  an  angle  *  different  from 
90°.  The  angle  e  between  the  horizon  and  the  initial  velocity  is 
called  the  elevation  of  the  projectile. 


88 


KINEMATICS. 


[167. 


Taking  the  horizontal  line  through  the  starting  point  O  as 
axis  of  x,  the  vertical  upwards  as  positive  axis  of  y  (Fig.  42),  the 


x 


Fig.  42. 

^-component  of  the  acceleration  is  evidently  o,  while  the  jj/-com- 
ponent  is  —  g\  hence,  by  (4),  Art.  160,  the  equations  of  motion 
are 

S=o>  §=-^-  (4> 

The  first  integration  gives 

dx  _£       dy_ 

As    -^=z/z,  -£ssyt   are  the  components  of  the  velocity  v  at 
at  at 

the  time  /,  the  constants  are  determined  from  the  values  of  vm  vv 
at  the  time  /=o;  viz.  VQ  cos  6=^,  VQ  sin  e  =  o  -f-  Cz.  We  have 
therefore 


(5) 


^z=  —  =  VQ  cos  e,  vy  =  -«  =  VQ  sin  e  —  gt. 

Integrating  again,  we  find 

X=VQ  cos  e  •  t,  y  —  ^Q  sin  e  •  t  —  \gfi,                 (6) 

the  constants  of  integration  being  o,  since  for  t=o  we  have 


These  values  of  x,  y,  vx,  vy  show  that  the  motion  in  the  hori- 
zontal direction  is  uniform  while  in  the  vertical  direction  it  is 


i68.] 


PLANE   MOTION. 


uniformly  accelerated.     This  is  otherwise  directly  evident  from 
the  nature  of  the  problem. 

Eliminating  t  between  the  expressions  for  x  and  y,  we  find 
the  equation  of  the  path 


cose 


(7) 


which  represents  a  parabola  passing  through  the  origin.  To 
find  its  vertex  and  latus  rectum,  divide  by  the  coefficient  of  x% 
and  rearrange  : 


2Vf 


•y\ 


completing  the  square  in  x,  the  equation  can  be  written  in  the 
form 

.2 

—  -^  sm  2  e 

*g  J  g 


Y=  -^cos2*  (y  -  ^sin?e\  (;' 

J  pr  V          2£  J 


V* 

ie    co-ordinates    of    the  vertex  are    therefore   <*=-^-sin2e, 

V  2  2  V  2  & 

?  =  —  sin2e;    the  latus  rectum  4  a  =  — -  cos2e ;   the  axis  is 

O  <3 

:ical,  and  the  directrix  is  a  horizontal  line  at  the  distance 

v  2 

—  cos2e  above  the  vertex. 


168.   Exercises. 


(1)  Show  that  the  velocity  at  any  time  is  v  =  V?'o2  —  2gy. 

(2)  Prove  that  the  velocity  of  the  projectile  is  equal  in  magnitude 
to  the  velocity  that  it  would  acquire  by  falling  from  the  directrix :  (a)  at 
the  starting  point,  (b)  at  any  point  of  the  path  (see  Art.  113). 

(3)  Show  that  a  body  projected  vertically  upwards  with  the  initial 
velocity  VQ  would  just  reach  the  common  directrix  of  all  the  parabolas 
described  by  bodies  projected  at  different  elevations  €  with  the  same 
initial  velocity  VQ. 


.90  KINEMATICS.  [168 

(4)  The  range  of  a  projectile  is  the  distance  from  the  starting  poin 
to  the  point  where  it  strikes  the  ground.     Show  that  on  a  horizonta 

v  2 
plane  the  range  is  R  =  2  a  =   °  sin  2  e. 

o 

(5)  The  time  of  flight  is  the  whole  time  from  the  beginning  of  the 
motion  to  the  instant  when  the  projectile  strikes  the    ground.     It   is 
best  found  by  considering  the  horizontal  motion  of  the  projectile  alonej 
which  is  uniform.     Show  that  on  a  horizontal  plane  the  time  of  flight  is 

7V=™°sine. 
g 

(6)  Show  that  the  time  of  flight  and  the  range  on  a  plane  through 
the  starting  point  inclined  at  an  angle  0  to  the  horizon  are 

r_2Vo  sin(c  —  6)       A    r,  _  2zy*  sin(e  —  0)cose 
j.  Q  —  --       —  -  —  ana  Ytg  —  —  — 

g       cos  0  g  cos  2  9 

(7)  What  elevation  gives  the  greatest  range  on  a  horizontal  plane? 

(8)  Show  that  on  a  plane  rising  at  an  angle  0  to  the  horizon,  to  obtain 
the  greatest  range,  the  direction  of  the  initial  velocity  should  bisect  the 
angle  between  the  plane  and  the  vertical. 

(9)  A  stone  is  dropped  from  a  balloon  which,  at  a  height  of  1000  ft., 
is  carried  along  by  a  horizontal  air-current  at  the  rate  of  15  miles  an 
hour,     (a)  Where,  (b)  when,  and  (c)  with  what  velocity  will  it  reach 
the  ground  ? 

(10)  What  must  be  the  initial  velocity  VQ  of  a  projectile  if  with  art 
elevation  of  30°  it  is  to  strike  an  object  100  ft.  above  the  horizontal 
plane  of  the  starting  point  at  a  horizontal  distance  from  the  latter  of 
1200  ft.? 

(n)  What  must  be  the  elevation  e  to  strike  an  object  100  ft.  above 
the  horizontal  plane  of  the  starting  point  and  5000  ft.  distant,  if  the 
initial  velocity  be  1200  ft.  per  second? 

(12)  Show  that  to  strike  an  object  situated  in  the  horizontal  plane  of 
the  starting  point  at  a  distance  x  from  the  latter,  the  elevation  must  be 
e  or  90°—  e,  where  €  =  ^sin 


(13)    The  initial  velocity  VQ  being  given  in  magnitude  and  direction, 
show  how  to  construct  the  path  graphically. 


i;o.]  PLANE   MOTION.  gx 

(14)  If  it  be  known  that  the  path  of  a  point  is  a  parabola  and  that 
the  acceleration  is  parallel  to  its  axis,  show  that  the  acceleration  is 
constant. 

(15)  Prove  that  a  projectile  whose  elevation  is  60°  rises  three  times 
as  high  as  when  its  elevation  is  30°,  the  magnitude  of  the  initial  velocity 
being  the  same  in  both  cases. 

(16)  Construct  the  hodograph  for  parabolic  motion,  taking  the  focus 
as  pole  and  drawing  the  radii  vectores  at  right  angles  to  the  velocities. 

169.  A  projectile  moving  in  the  air  or  in  any  other  resisting 
medium  of  uniform  density  is  subject,  in  addition  to  the  con- 
stant acceleration  g  of  gravity,  to  the  resistance  of  the  medium 
which  produces  a  retardation  variable  both  in  magnitude  and 
direction  (Art.  125).     Experiment  shows  that  this  retardation 
jean  he  expressed  in  the  form  cvn,  where  c  is  constant  for  a  given 
projectile  and  medium,  and  n  must  be  determined  by  experiment 
for  different  initial  velocities. 

170.  For   n=i    the   integrations    can    be    readily   effected. 
Resolving  tbe  retardation  cv  into  its  components  cvx=cdxjdt, 

y—cdyjdt,  the  equations  of  motion  are 


d^x  dx     d*y  dy 

-=~c          -c 


Integrating,  we  find 

^=^  =  ^0  cos  €•*-",    vy  =  ^-=l-[-g+(cv^me+g)e-ct],  (9) 
ctt  dt     c 

since  for  t—Q  we  have  vx  =  v$  cose,  z/y  =  z/0  sin  e. 
The  second  integration  gives 

(10) 


92 


KINEMATICS. 


since  #=0,  jj/=o  for  t—O.     Eliminating  t,  we  find  the  equation! 
of  the  path  of  the  projectile : 


,0)86  —  CX 


Vn COS  6 


(II) 


The  curve  has  a  vertical  asymptote  x=-  —      — ;  for  this  value 
of  x,  /=oo . 

171.    Uniform  Circular  Motion.    Let  a  point  P  (Fig.  43)  describe 
a  circle  of  radius  a  with  constant  angular  velocity  co.     Its  linear 


Fig.  43. 

velocity  v  =  wa  is  of  constant  magnitude,  but  varies  in  direction. 
By  the  formulae  (i),  (2)  of  Art.  159,  the  tangential  acceleration 
jt  =  o,  while  the  normal  acceleration  jn  =  v2'/a  =  aPa  represents 
the  total  acceleration.  Hence,  in  uniform  circular  motion,  the 
acceleration  is 

j=*  =  tfat  (12) 


and  is  always  directed  toward  the  centre  O  of  the  circle. 

This  appears  also  by  constructing  the  hodograph  of  the 
motion,  which  is  evidently  a  circle  of  radius  v  (Fig.  43).  As 
the  acceleration  of  P  is  represented  by  the  velocity  of  the  point 
P'  of  the  hodograph  (see  Art.  156),  we  have  only  to  determine 
this  velocity.  Let  T  be  the  so-called  period,  or  periodic  time, 
i.e.  the  time  in  which  the  point  P  makes  a  whole  revolution, 


730 


PLANE   MOTION. 


93 


;o  that  T=2Tra/v\  then,  since  P'  describes  the  circle  of  radius 
in  the  same  time  T,  we  have  for  the  velocity  of  P'  the  expres- 
>ion  2TTV/T,  or  substituting  for  T  its  value,  v*/a,  as  above. 

172. .  Simple  harmonic  motion  is  a  rectilinear  motion  in  which 
he  distance  x  of  the  moving  point  Px  (Fig.  44)  from  a  fixed 
>rigin  O  in  the  line  of  motion  is  a  simple  harmonic  function  of 
he  time,  i.e.  a  function  of  the  form 

vhere  a,  o>,  e  are  constants. 

If  the  positions  P  of  a  point  moving  uniformly  in  a  circle  be 
)rojected  at  every  instant  on  any  diameter  A  A'  of  the  circle,  it 
s  easy  to  see  that  the  motion 
)f  the  projection  Px  along  the 
liameter  is  simply  harmonic.  For 
lenoting  the  constant  angular 
elocity  of  P  by  &>,  the  angle 
4OP  will  be  =  ft>/  if  the  time  be 
:ounted  from  the  point  A.  Hence 
he  distance  OPx=x  of  the  point 
Px  from  the  centre  O,  or  the  dis- 
)lacement  of  Px  at  the  time  ty  is 

;r=tfCOSG)/,  Fig.  44. 

vhere  a  is  the  radius  of  the  circle.      This  radius  a=OA  is 
:alled  the  amplitude  of  the  simple  harmonic  motion. 

173.  While  P  moves  uniformly  in  the  circle,  its  projection 
»  evidently  performs  oscillations  from  A  through  O  to  A'  and 
3ack  through  O  to  A. 

The  time  T  of  completing  one  whole  oscillation  forward  and 
Backward  is  called  the  period  of  the  simple  harmonic  motion  ; 
t  is  obviously  equal  to  the  period  of  the  motion  of  P  in  the 
:ircle ;  i.e. 

T=2-^-  (14) 


KINEMATICS.  [ 

94 

The  period  is  therefore  independent  of  the  amplitude  a.     Il 
follows  that  two  simple  harmonic  motions  resulting  from  tw< 
uniform  circular  motions  of  the  same  angular  velocity  on  two  I 
concentric  circles  of  different  radii  have  the  same  period  ;  such] 
motions  are  called  isochronous. 

174.  If  the  time  /  be  counted,  not  from  A,  but  from  some 
other  point  PQ  on  the  circle  for  which  %AOPQ  =  e,  we  have 
^AOP=(0t+e,  and  the  equation  of  the  simple  harmonic 

motion  is 

05) 


The  angle  o>/+e  is  called  the  phase-angle,  while  e  is  the  epoch-j 
angle,  of  the  motion.  The  names  phase  and  epoch  are  sometimesj 
applied  to  these  angles,  although,  strictly  speaking,  the  phase  is 
the  time  (usually  expressed  as  a  fraction  of  the  period  T)  oi\ 
passing  from  the  position  A  of  maximum  displacement  to  any 
position  P*  while  the  epoch  is  the  phase  corresponding  to  the 
time  t=o. 

175.   Differentiating  equation  (15),  we  find  the  velocity 

vx  =  —  =  -aco  sin  (orf+e)  ;  (i6)l 

at 

and  differentiating  again,  we  obtain  the  acceleration 

(I?) 


of  simple  harmonic  motion. 

The  same  values  can  be  derived  by  projecting  the  velocity 
and  acceleration  of  the  uniform  circular  motion  of  P  on  the 
diameter  A  A  ',  as  is  readily  seen-  from  Fig.  44. 

176.  Equation  (17)  shows  that  in  simple  harmonic  motion  the 
acceleration  is  directly  proportional  to  the  distance  from  the 
centre. 


1770 


PLANE   MOTION. 


95 


Conversely,  it  can  be  shown  that  if  the  acceleration  be  pro- 
portional to  the  distance  from  a  fixed  point  in  the  direction  of 
the  initial  velocity,  and  if  it  be  directed  towards  this  point,  the 
motion  is  simply  harmonic.  For  we  then  have 


•ar 


dt* 

where  fi  is  constant.     The  general  integral  of  this  differential 
equation  is  (compare  Art.  122) 


x= 


sn 


C2  cos 


Differentiating,  we  find  for  the  velocity 

v=  C^i  cos  i^t—  C^fi  sin  fit. 

To  determine  the  constants  of  integration  Cv  C^  let  s=s^ 
and  V  =  VQ  at  the  time  /=o.  Substituting  these  values,  we  find 
r0=6*2  and  v^fiC^  hence 

x=  ^  sin  fit+  s0  cos  fit. 
1* 

Putting  vQ/fi=acose,  sQ=a  sin  e,  which  is  always  allowable,. 

|ve  obtain 

x=a  (sin  fit  cos  e+  cos  fit  sin  e) 


—a  sn 
This  represents  a  simple  harmonic  motion  whose  amplitude 


2/yLt,  and  whose  epoch-angle  is  e=  tan'1^/^). 
is  the  angular  velocity  of  the  corresponding  uniform  circular 
potion  is  //,,  the  period  is  T—  2  TT//^. 

177.    If  the  uniform  circular  motion  of  P  be  projected  on  the 

[ameter  BB',  which  is  at  right  angles  to  the  diameter  A  A' 

'ig.  44),  we  have  OPy=y  =  a  sin  (o)/+e).     Writing  this  in  the 

liuivalent  form 

f  Tr\ 

y—  —a  cos  f  w/H-eH  —  J, 


KINEMATICS. 


[178. 


it  appears  that  the  motion  of  Py  is  simply  harmonic  of  the  same 
period  and  amplitude  with  the  motion  of  Px,  but  differing  by 
-7T/2  in  phase. 

178.  Simple    harmonic    motions    occur    very   frequently    in 
applied   mechanics    and    mathematical    physics.     A    particular 
case  has  been  treated  in  Arts.  121-124.     As  another  example 
we  may  mention  the  apparent  motion  of  a  satellite  about  its 
primary  as  seen  from  any  point  in  the  plane  of  the  motion,  pro- 
vided the  satellite  be  regarded  as  moving  uniformly  in  a  circle 
relatively  to  its  primary.     Thus  the  moons  of  Jupiter,  as  seen 
from  the  earth,  have  approximately  a  simple  harmonic  motion. 

179.  A  mechanism  for  producing  simple  harmonic  motion 
can  readily  be  constructed  as  follows.     The  end  A   (Fig.  45) 

of  a  crank  rotating  uniformly 
about  the  axis  O,  carries  a  pin 
running  in  the  slot  AB  of  a  T 
bar  ABD  whose  axis  (produced)  j 
passes  through  the  centre  O  of! 
the  crank  circle.  The  T  bar  is 
constrained  by  guides  to  move 
back  and  forth  along  the  line  OD ;  its  motion  is  evidently  simply! 
harmonic,  the  uniform  circular  motion  of  the  crank  being  trans-l 
formed  into  rectilinear  motion.  Compare  Art.  151. 

180.  Exercises. 

(i)  Show  that  the  maximum  acceleration  of  the  simple  harmonic 
motion  is  numerically  equal  to  the  acceleration  in  the  corresponding 
uniform  circular  motion. 

.    (2)   Find  the  time   of  one  oscillation  from   equation  (15)  without 
reference  to  the  circular  motion. 

(3)  In  the  mechanism  shown  in  Fig.  45,  if  the  length  of  the  crank 
2  feet  and  the  number  of  revolutions  15  per  minute,  find  the  velocity^ 
.and  acceleration  of  the  end  D  of  the  T  bar :  (a)  when  at  elongation ; 
.(£)  when  at  quarter  stroke  ;  (<r)  when  at  the  middle  of  the  stroke. 


182.] 


PLANE   MOTION. 


97 


(4)  Show  that  the  period  of  a  simple  harmonic  oscillation  can  be 
•expressed  in  the  form  T=  2  TT  V  —  x/jx  where  jx  is  the  acceleration  of 
the  oscillating  point  at  the  time  when  its  distance  from  the  centre,  or 
its  displacement,  is  x. 

(5)  Px,  P'x  being  the  positions  of  the  oscillating  point  at  the  times 
/,  /',  respectively,  and  8  the  angle  POP,  i.e.  the  difference  of  phase, 
.show  that  f  —  /=8/w. 

(6)  Show  that  vx  =  —  o>  V#2  —  x?. 


181.  Compound  Harmonic  Motion.      We  have  seen  (Art.  176) 
that  the  motion  of  a  point,  whose  acceleration  is  directly  propor- 
tional to  its  distance  from  a  fixed  centre,  and  directed  towards 
this  centre,  is  simply  harmonic,  provided  the  centre  lies  in  the 
line  of  the  initial  velocity.     Removing  this  last  restriction,  we 
have  the  more  generalise  of  compound  harmonic  motion. 

Let  O  (Fig.  46)  be  the  centre,  P  the  position  of  the  moving 
point  at  the  time  t,  OP  —  s  its  distance  from  the  centre,  v  its 
velocity,  j=  —/j?s  its  accelera- 
tion, at  that  time.  Referring 
the  motion  to  two  rectangular 
axes  Ox,  Oy  in  the  plane  deter- 
mined by  v  and  O,  we  can 
resolve  v  and  j  into  their  com- 
ponents along  these  axes : 

vx  =  v  cos  «,   vy  =  v  sin  a, 
and  jx=  —fix,  jy=  —^2y,  where 
a  is  the  angle  made  by  v  with  the  axis  Ox,  and  x,  y  are  the 
co-ordinates  of  P. 

The  projections  Px,  Py  of  P  on  the  axes  have  therefore  each 
a  simple  harmonic  motion,  and  the  motion  of  P  may  be  regarded 
as  the  resultant  of  these  component  motions. 

182.  In  general,  the  motion  of  P  will  be  curvilinear.     We 
proceed  to  examine  somewhat  more  in  detail  the  most  important 
cases  of  the  composition  of  two  or  more  simple  harmonic  motions, 

PART   I — 7 


9g  KINEMATICS.  [183. 

beginning  with  those  cases  in  which  the  resultant  motion  is 
rectilinear. 

As,  according  to  Hooke's  law,  the  particles  of  elastic  bodies, 
after  release  from  strain  within  the  elastic  limits,  perform  small 
oscillations  for  which  the  acceleration  is  proportional  to  the 
displacement  from  a  middle  position,  the  motions  under  discus- 
sion find  a  wide  application  in  the  theories  of  elasticity,  sound,, 
light,  and  electricity,  and  form  the  basis  of  the  general  theory 
of  wave  motion  in  an  elastic  medium. 

183.  Two  simple  harmonic  motions  in  the  same  line,  of  equal 
amplitude  a  and  equal  period  T,  but  differing  in  phase  by  &,  com- 
pound into  a  simple  harmonic  motion  in  the  same  line,  of  the- 
same  period  T,  but  having  the  amplitude  2  a  cos  (8/2). 

For  we  have  for  the  component  displacements 

x^=a  cos  co/,  x^  —  ft  cos  (fr>^+S) ; 

and  as  these  are  in  the  same  line,  they  can  be  added  algebrai- 
cally giving  the  resultant  displacement 


=  #[cos  o)/-hcos(i 
or,  by  the  formula  cos  a  -f  cos  0  =  2  cos  "     ^  cos 


a  — 


x—  2  a  cos-  •  cos    &>/  +  -. 

2  V  2 

184.  Two  simple  harmonic  motions  in  the  same  line,  of  equal 
period  T,  but  differing  both  in  amplitude  and  in  phase,  compound 
into  a  single  simple  harmonic  motion  in  the  same  line  and  of  the 
same  period. 

For  the  component  displacements 


^1  =  al  cos  w/H-ej,  ^2  =  <z2  cos 

can  again  be  added  algebraically,  and  the  resultant  displacement 
is 


i8s-] 


PLANE   MOTION. 


a1cos  (wt  +  e^)+  tf2  cos  (at  +  e2) 
=  (tfj  cos  ej-h  <z2  cos  e2)  cos  w/—  (tfj  sin  ejH-^  sin  e2)  sin  at. 

Putting   <21cose1  +  ^2cose2  =  ^cose,      a±  sine^d^  sine2  =  0  sin  e, 
we  have 

x=a  cos  e  cos  a>t—a  sin  e  sin  a*t 

—  a  cos  (o)/+e), 
where   a2  =  (#  j  cos  e1  +  ^2  cos  e2)2  +  (a^  sin  6a 

=  aj2  H-  «22  +  2  tf1(z2  cos  (ea  -  e^ 
and  tan  e=  (a-^  sin  e-^  +  a^  sin  e^)/(a1  cos  ej-f-tfg  cos  62)- 

185.  A  geometrical  illustration  of  the  preceding  proposition 
is  obtained  by  considering  the  uniform  circular  motions  corre- 
sponding to  the  simple  harmonic  motions  (Fig.  47). 


sin  e2) 


Fig.  47. 


Drawin    the  radii 


=  tf  so  as  to  include  an  angle 


equal  to  the  difference  of  phase  e2  —  €1  and  completing  the 
parallelogram  OP^PP^  it  appears  from  the  figure  that  the 
diagonal  OP  of  this  parallelogram  represents  the  resulting 
amplitude  a.  For  since  P^P  is  equal  and  parallel  to  OP^  we 
have  for  the  projections  on  Ox  the  relation  OPx^+OP*t=OPx, 
or  x^+x^=x. 

Again,  if  the  angle  xOPl  be  taken  equal  to  the  epoch-angle 
ep  and  hence  ^OP^  =  e2,  the  angle  xOP  represents  the  epoch 
e  of  the  resulting  motion. 


100 


KINEMATICS. 


[186. 


AQ 


We  thus  have  a  simple  geometrical  construction  for  the 
elements  a,  e  of  the  resulting  motion  from  the  elements  alt  e1 
and  «2,  e2  of  the  component  motions.  As  the  period  is  the 
same  for  the  two  component  motions,  the  points  Pl  and  P2 
describe  their  respective  circles  with  equal  angular  velocity  so 
that  the  parallelogram  OP^PP^  does  not  change  its  form  in  the 
course  of  the  motion. 

186.  The  construction  given  in  the  preceding  article  can  be 
described  briefly  by  saying  that  two  simple  harmonic  motions 
of  equal  period  in  the  same  line  are  compounded  by  geometrically 
adding  their  amplitudes,  it  being  understood  that  the  phase- 
angles  determine  the  directions  in  which  the  amplitudes  are  to 
be  drawn. 

It  follows  at  once  that  not  only  two,  but  any  number  of  simple 
harmonic  motions,  of  equal  period  in  the  same  line,  can  be  com- 
pounded by  geometric  addition 
of  their  amplitudes  into  a  sin- 
gle simple  harmonic  motion  in 
the  same  line  and  of  the  same 
period. 

Conversely,  any  given  sim- 
ple harmonic  motion  can  be 
F  resolved    into    two   or   more 
components  in  the  same  line 
and  of  the  same  period. 

187.  The  kinematical  mean- 
ing of  this  composition  of  sim- 
ple harmonic  motions  of  equal 
period  in  the  same  line  will 
perhaps  be  best  understood 
from  the  mechanism  sketched 
in  Fig.  48.  A  cord  runs  from 

the  fixed  point  A  over  the  movable  pulleys  B,  D  and  the  fixed 
pulleys  C,  E,  and  ends  in  F.      Each  of  the  movable  pulleys 


Fig.  48. 


1 88.]  PLANE   MOTION. 


lor 


receives  a  vertical  simple  harmonic  motion  from  the  T  bars  BG 
and  DH,  just  as  in  Fig.  45  (Art.  179).  If  the  free  end  F  of  the 
cord  be  just  kept  tight,  its  vertical  displacement  will  be  twice 
the  sum  of  the  vertical  displacements  of  B  and  D,  and  as  these 
points  have  simple  harmonic  motions,  the  motion  of  F  will  be 
twice  the  resultant  simple  harmonic  motion. 

The  idea  of  this  mechanism  is  due  to  Lord  Kelvin. 

188.   Exercises. 

(  i  )  Find  the  resultant  of  three  simple  harmonic  motions  in  the  same 
line,  and  all  of  period  T=  10  seconds,  the  amplitudes  being  5,  3,  and 
4  cm.,  and  the  phase  differences  30°  and  60°,  respectively,  between  the 
first  and  second,  and  the  first  and  third  motions. 

(2)  Apply   the   geometrical   method  of  Art.    185    to   the  problem 
of  Art.  183. 

(3)  Find  the  resultant  of  two  simple  harmonic  motions  in  the  same 
line  and  of  equal  period  when  the  amplitudes  are  equal  and  the  phases 
differ  :   (a)  by  an  even  multiple  of  TT,  (£)  by  an  odd  multiple  of  TT. 

(4)  Resolve  x  =  10  cos  (TT/+  45°)  into  two  components  in  the  same 
line  with  a  phase  difference  of  30°,  one  of  the  components  having  the 
epoch  o. 

(5)  Trace  the  curves  representing  the  component  motions  as  well  as 
the  resultant  motion  in  Ex.  (i),  taking  the  time  as  abscissa  and  the 
displacement  as  ordinate. 

(6)  Show  that  the  resultant  of  n  simple  harmonic  motions  of  equal 
period  Z'in  the  same  line,  viz.  : 

^/4-  A  .-  *„  =  0nCos 
is  the  isochronous  simple  harmonic  motion 


==  *!  cos  *2  = 


x  =  a  coSi   T 


where  a*  =  (2at  cos  Ci)2  +  (%*t  sin  * 


*_L_ /"?./»    cin  t.\2 


and  tan  c  =  20<  sin  e^/Stf,  cos  «<. 


I02  KINEMATICS.  [189. 

189.  The  composition  of  two  or  more  simple  harmonic  motions 
in  the  same  line  can  readily  be  effected,  even  when  the  compo- 
nents differ  in  period.  But  the  resultant  motion  is  not  simply 
harmonic. 

Thus,  for  two  components 

#2  =  02  cos(ft>2/-he2), 


putting  w2^+e2  =  «1;+(«2-&)i)/  +  e2  =  &)i/+€i  +  ^  Sa7»  where 
S=(<»2  —  (»!)/+  €2  —  6!  is  the  difference  of  phase  at  the  time  ty  we 
have  for  the  resulting  motion 


=al  cos 
and  treating  this  similarly  as  in  Art.  184  we  find 

x—(al-\-  a2  cos  8)  cos  (o>^  +  ^)  —  #2  sin  8  sin 
or  putting  a1-{-a2  cos  8  =  a  cose,     a%  sin  8=  a  sine, 

x  =  a  cos  («!/  +  ej  +  e), 
where  a2  =a12  +  a%2  +  2afy  cos  8  and  tane=^2  sin8/(a1+a2  cos 


190.  These  formulae  can  be  interpreted  geometrically  by 
Fig.  47,  similarly  as  in  Art.  185.  But  as  in  the  present  case 
the  angle  8,  and  consequently  the  quantities  a  and  e  in  the 
expression  for  x,  are  variable,  the  parallelogram  OP1PP2  while 
having  constant  sides  has  variable  angles  and  changes  its  form 
in  the  course  of  the  motion. 

A  mechanism  similar  to  that  of  Fig.  48  (Art.  187),  can  be 
used  to  effect  mechanically  the  composition  of  simple  harmonic 
motions  in  the  same  line  whether  the  periods  be  equal  or  not. 
This  is  the  principle  of  the  tide-predicting  machine  devised  by 
Lord  Kelvin.* 


*  See  THOMSON  and  TAIT,  Natural  philosophy,  Vol.  I.,  Part  I.,  new  edition,  1879, 
p.  43  sq.  and  p.  479  sq.  and  J.  D.  EVERETT,  Vibratory  motion  and  sound,  1882. 


I92.j 


PLANE   MOTION. 


103 


191.  To  show  the  connection  of  the  present  subject  with  the 
theory  of  wave  motion,  imagine  a  flexible  cord  AB  of  which  one 
end  B  is  fixed  while  the  other  A  is  given  a  sudden  jerk  or 
transverse  motion  from  A  to  C  and  back  through  A  to  D,  etc. 
(Fig.  49).  The  displacement  given  to  A  will,  so  to  speak,  run 
along  the  cord,  travelling  from  A  to  B  and  producing  a  wave. 
The  figure  exhibits  the  successive  stages  of  the  motion  up  to 
the  time  when  a  complete  wave  has  been  produced. 


A' 


Fig.  49. 


192.  The  distance  A'K  (Fig.  49)  is  called  the  length  of  the 
wave.  Denoting  this  length  by  X,  and  the  time  in  which  the 
motion  spreads  from  A '  to  K  by  T  we  have  for  the  velocity  of 
propagation  of  the  wave 

V=j:  (18) 

It  is  to  be  noticed  that  the  motion  of  any  particular  point  of 
the  cord  is  supposed  to  be  rectilinear  and  at  right  angles  to 
AB ;  this  is  the  case  with  the  simple  transverse  vibrations 
in  an  elastic  medium  such  as  the  luminiferous  ether  regarded 
as  the  vehicle  of  light  waves. 


104  KINEMATICS.  [193. 


193.  If  the  motion  of  A  be  simply  harmonic,  sayjy  =  # 

the  motions  of  the  successive  points  of  the  cord  will  differ  from 
the  motion  of  A  only  in  phase,  and  the  displacements  of  all 
these  points  at  any  time  t  can  be  represented  by 

y  =  a  sin(W—  e),  (19) 

where  e  varies  from  o  to  2  TT  as  we  pass  from  A  to  K. 

As  the  time  T  in  which  the  motion  spreads  from  A  to  K  is 
equal  to  the  period  of  a  vibration  of  A  (or  of  any  other  point  of 
the  cord),  we  have  co  =  27r/T,  or,  by  (18),  o)  =  27rF/X.  And  if  x 
be  the  distance  of  the  point  of  the  cord  under  consideration  from 
A,  we  must  have  x  :  \  =  e  :  2?r  ;  that  is,  e  =  27rx/\.  Substituting- 
these  values  of  co  and  e  in  (19),  the  equation  of  the  wave  motion 
can  be  written  in  the  form 

y=a*\TL—(Vt-x).  (20) 

X 

194.  This  equation  can  be  looked  upon  from  two  different 
points  of  view  according  as  we  regard  /  or  x  as  variable. 

Let  /  be  constant  ;  i.e.  let  us  consider  the  displacements  of 
all  points  of  the  cord  at  a  given  instant.  If  for  x  in  (20)  we 
substitute  x+n\,  where  n  is  any  positive  or  negative  integer, 
the  angle  (Vt—  x}  2ir/\  is  changed  by  2irn,  so  that  the  value 
of  y  remains  unchanged.  The  displacements  of  all  particles 
whose  distances  from  A  differ  by  whole  wave  lengths  are  there- 
fore the  same;  in  other  words,  the  state  of  motion  at  any 
instant  is  represented  by  a  series  of  equal  waves. 

Now  let  x  be  constant,  and  t  variable.  Substituting  for  t  in 
(20)  the  value  t+nT=t+n\/V,  the  angle  (Vt—x)  2?r/X  is  again 
changed  by  27r«,  and  y  remains  the  same.  This  shows  the 
periodicity  in  the  motion  of  any  given  particle. 

195.  If  the  point  A  (Fig.  49)  be  subjected  simultaneously  to 
more   than   one   simple   harmonic    motion,    the   displacements 
resulting  from  each  must  be  added  algebraically,  thus  forming- 
a  compound  wave  which  can  readily  be  traced  by  first  tracing 


I9S.J  PLANE   MOTION.  IC>5 

the  component  waves  and  then  adding  their  ordinates,  or  ana- 
lytically by  forming  the  equation  of  the  resultant  motion  as  in 
Art.  189. 

196.  Exercise. 

(i)  Trace  the  wave  produced  by  the  superposition  of  two  simple 
harmonic  motions  in  the  same  line  of  equal  amplitudes,  the  periods 
being  as  2:1,  (a)  when  they  do  not  differ  in  phase,  (b)  when  their 
epochs  differ  by  7/16  of  the  period. 

197.  The  idea  of  wave  motion  implies  that  the  displacement 
y  should  be  a  periodic  function  of  x  and  t  such  as    to   fulfil 
the  following  conditions  :  y  must  assume  the  same  value  (a) 
when  x  is  changed  into  n\,  (&)  when  t  is  changed  into  /+  T, 
(c)  when  -both  changes  are  made  simultaneously;  the  constants 
\  and  T  being  connected  by  the  relation  \=  VT. 

The  condition  (c)  requires  y  to  be  of  the  form  y—f(Vt—x]  ; 
for  Vt—  x  remains  unchanged  when  x  is  replaced  by  x-\-  VT 
and  at  the  same  time  /by  /+  T. 

A  particular  case  of  such  a  function  is  y=a  smc(Vt—x).  As 
y  should  remain  unchanged  when  /  is  replaced  by  /+  Tt  we 
must  have  c=2Tr/VT=2Tr/\.  Thus  the  function 

7=^  sin  —  (Vt-x) 
X 

fulfils  the  three  conditions  (a),  (b),  (c).  Putting  as  before  (Art. 
193)  2irx/\  =  —  e,  we  can  write  it 


198.  The  importance  of  this  particular  solution  of  our  problem  lies 
in  the  fact  that,  according  to  Fourier's  theorem,  any  single-valued 
periodic  function  of  period  T  can  be  expanded,  between  definite  limits 
of  the  variable,  into  a  series  of  the  form 


(21) 


100  KINEMATICS.  [199. 

As  applied  to  the  theory  of  wave  motion  this  means  of  course  that  any 
'wave  motion,  however  complex,  can  be  regarded  as  nude  up  of  a  series 
of  superposed  simple  harmonic  vibrations  of  periods  Tt  T/2,  7/3,  •••, 
or  since  T=  K/V,  of  wave  lengths  A,  A/a,  A/3,  .... 

199.  A  full  discussion  of  Fourier's  theorem  cannot  be  given  in  this 
place.0    We  wish,  however,  to  show  its  practical  application  in  an 

example. 
The  equation  (ai)  can  be  written  in  the  form 

/(/)  =  «!  cosej  sin.  /+  a*  cos  e8  sin.  a  /+  <*3  cos  e3  sin-  3/+  »•• 


or  putting    a  irfT'ss  #,  <*!  sin  €j 
02  cos  ea  =  /?a,  •  ••, 


a0  +  -^  cos  x  -f  At  cos  *x  +  ^8  cos  3*  +  — 

-f  ^sin^e  -f^asin  2x  4-^?3sin3Jc  4-  •••.         (aa) 


This  is  known  as  Fourier's  series.  According  to  the  nature  of  the  func- 
tion to  be  expanded,  it  is  often  sufficient  to  use  the  sine  series  or  the 
cosine  series  alone.  As  the  method  of  determining  the  coefficients  is 
always  the  same,  it  will  be  sufficient  to  consider  the  simple  sine  series  : 


200.  The  problem  before  us  can  now  be  stated  as  follows  :     Given  any 

single-valued  function  of  je,  either  by  its  analytical  expression  or  by  the 

trace  of  the  curve  representing  it,  to  determine  the  coefficients  B  it 

(33)  so  as  to  make  the  right-hand  member  of  this  equation  represenf 

the  values  of  the  given  function  between  certain  finite  limits  of  x. 

.  We  shall  assume  these  limits  to  be  x  =  o  and  x  =  *•  :  and  we  shall 


*The  student  is  referred  to  THOMSON  and  TAIT,  Natural  philosophy*  I.  I,  1879, 
pp.  55-60;  also  to  B.  RIBMANN,  Partielk  Di/trtHtialgieickHHgv*%  herausgegeben  votf 
K.  Hattendorff,  3d  ed.,  Braunschweig,  Vieweg,  1882,  pp.  44-95,  and  to  G.  M. 
MINCHIN,  L-ttiplanar  tintma(tcs,  Oxford,  Clarendon  Press,  1882,  p.  13  sq. 


202.]  PLANE   MOTION.  IO; 

first  try  to  make  n  —  i  points  of  the  given  curve  taken  between  these 
limits  coincide  with  as  many  points  of  the  curve 


f(x)  =  Z?!  sin x  +  BI  sin  2 ^  H +Bn  !  sin  («  —  i)x.         (24) 

Then  passing  to  the  limit  for  «  =  oc,  the  problem  will  be  solved. 

201.  Dividing  the  interval  from  x  =  o  to  x  =  ir  into  «  equal  parts 
and  taking  the  «  —  i  points  on  the  given  curve  so  that  their  abscissae 
are 

TT    2ir    37T         (rt—  I)T 

n     n      n  n 

he  curve  (24)  will  contain  these  points  if  the  following  n  —  i  equations 
are  fulfilled : 


fl  J "    \  n       .  "  .     r»      .       _      2  7T 

_.  .  ^ 


=  ^lSin(»-i)^+^sin(«-i)^+...4-^n-isin(«-i)^-^. 

.     .     .     (25) 
These  equations  are  sufficient  to  determine  the  «— i  unknown  con- 


stants  3lt  B*  .»,  ^n_i. 

202.  To  solve  the  equations  we  multiply  them  by  indeterminate  co- 
efficients and  add.  The  coefficients  can  be  so  selected  as  to  make  all 
the  unknown  quantities  disappear  except  one  which  is  thus  determined. 
Thus,  to  find  Bm  multiply  each  equation  by  twice  the  coefficient  of 
Bm  in  this  equation,  viz.  the  first  equation  by  2  sin  (mir/ri),  the  second 
>y  2  sin (2  mir/n),  etc.,  the  last  by  2  sin[(«  —  i)mir/ri]. 
|  After  adding,  the  factor  of  Bk  will  be 

,  7T    .  7T  .  2  7T    .  2  7T    , 

Bk  =  2\  sm/fc-smw     +  sm£  —  smm h  ••• 

n          n  n  n 

+  sin  k  (n  —  i)  -sin  m(n  —  i)  -  • 


108  KINEMATICS.  [202. 

Transforming   every   term   by  the   formula   2  sin  a  sin  fi=  cos  (a—  /?) 
—  cos  (a+(3),  we  find 


.  .  .  (26) 

Applying  the  trigonometrical  formula 


(2;?-I)Ct/2\ 

sin  a/2        y 


we  obtain 


sin(2»  —  i)  (k  —  m)  —       sin(2  n  —  i)  (k  +  m)  — 
2      =  -  -  ---  -  -  — 


cos(£  —  w)7rsin(^  —  m)— 


(k  4-  ni\  — cos(£  4-  m\it sin(/&  +  m\  v 

V         ' 2n         *         x71"      v         ;^g 


If  £  be  different  from  tn,  this  reduces  to 


and  this  is  always  =  o,  since  k  +  m  and  k  —  m  are  either  both  odd 
or  both  even. 

If  k=m,  we  find  from  (26) 


[7T    .  27T    ,  («  —  iWH 

COS  2m-+  COS  2*#  --  1  ----  +  COS2*«^  -  '— 
n  n  n       J» 


:o4-] 


-1 


PLANE    MOTION. 

.       x  v    WITT 

n 


sin 


mir 


m-rr  .    mir 

sin  2  m-rr  cos cos  2  tmr  sin — • 

ii  n  n 

n 

2      2  .   WTT 

sm — 


109 


We  have,  therefore,  finally 


:kth  m  =  i,  2,  3,  •«•  n  —  i. 

203.    It  remains  to  p 
Voting  (27)  in  the  form 


27T 


f 
2/f 


f(n—  I)TT\    .         («—  I) 
f  ^—  —  ^-Jsin  «f^__2 


/      x 
(27) 


203.    It  remains  to  pass  to  the  limit  when  n  =  oo  and  —  vanishes. 


e  obviously  have  in  the  limit 

2    C* 

Bm=-  I   / 

TTc/O 


2,  3, 


(28) 


Fig.  50. 

204.  As  an  application  let  us  determine  the  series  representing  the 
roken  line  formed  by  the  two  sides  of  an  isosceles  right-angled  triangle 
rhose  hypotenuse  lies  in  the  axis  of  x  (Fig.  50). 


IIO  KINEMATICS.  [205' 

We  assume  the  length  of  this  hypotenuse  =TT;  then  the  given  funcj 
tion  isf(x)=x  from  x=o  to  x=ir/2}  and  /(X)=TT— x  from  x=Tr/2 
to  X=TT. 

On  account  of  the  discontinuity  at   the  point  x=ir/2,  the  Integra 
in  (28)  must  be  resolved  into  two,  and  we  have 

2  r  c^  c*  ~\ 

Bm=  -     I  x  sin  mx  dx  +  I  (ir—x)  sin  mx  dx 

7T  \_J  0  Jn  J 

2  f  I    7T  MITT  I        .      KlTT  I    TT  #27T  I       .      W7T~] 

= cos 1 — 5  sm 1 cos h  — 9  sin  — 

7T  L       m  2  2          #T  2          W  2  2  #T  2J 


For  even  values  of  m,  sin(w7r/2)=o  ;  for  odd  values,  sin(w7r/2)  is 
alternately  positive  or  negative.     Hence  the  series  (23)  becomes 


sin 

(29) 


This  expansion  certainly  holds  when  x  lies  between  o  and  ?r.  A 
every  term  of  the  series  vanishes  for  x=o  as  well  as  for  X=TT,  th< 
expansion  holds  even  at  these  limits.  Moreover,  when  x  lies  between  i 
and  2  TT,  all  the  terms  of  the  series,  with  signs  reversed,  pass  through  thi 
same  succession  of  values  as  between  o  and  TT.  The  series  represents 
therefore  between  these  limits  an  equal  triangle  with  its  vertex  belo^ 
the  axis  of  x  (Fig.  50).  Beyond  the  point  x=2-n,  the  same  figure 
repeats  itself  owing  to  the  periodicity  of  the  sine. 

It  thus  appears  that  the  series  represents  an  infinite  zigzag  line  fo 
all  values  of  x. 

205.  We  proceed  to  the  composition  of  simple  harmonic  motion* 
not  in  the  same  line.  We  shall,  however,  assume  that  all  tht 
component  motions  lie  in  the  same  plane. 

It  is  evident  that  the  projection  of  a  simple  harmonic  motion 
on  any  line  is  again  a  simple  harmonic  motion  of  the  same 
period  and  phase  and  with  an  amplitude  equal  to  the  projection 
of  the  original  amplitude. 


2o6.]  PLANE   MOTION.  Iir 

Hence,  to  compound  any  number  of  simple  harmonic  motions 
along  lines  lying  in  the  same  plane,  we  may  project  all  these 
motions  on  any  two  rectangular  axes  Ox,  Oy  taken  in  this 
plane,  and  compound,  by  Art.  184  or  189,  the  components  lying 
in  the  same  axis.  It  then  only  remains  to  compound  the 
two  motions,  one  along  Ox,  the  other  along  Oy,  into  a  single 
motion. 

206.  Just  as  in  Arts.  184,  189,  we  must  distinguish  two 
cases  :  (a)  when  the  given  motions  have  all  the  same  period, 
and  (f)  when  they  have  not. 

In  the  former  case,  by  Art.  184,  the  two  components 
along  Ox  and  Oy  will  have  equal  periods,  i.e.  they  will  be  of  the 
form 

x—a  sin  &>/,    y  =  b  sin  (&>/+£).  (30) 

The  path  of  the  resulting  motion  is  obtained  by  eliminating  t 
between  these  equations.  We  have 


•^=  sinft>£cos£-h  cos  wt  sin  8 
o 


x  I *2 

=-  cosS+\/i  — «  sin 8. 

/7  »  SI" 


Writing  this  equation  in  the  form 


or  5-^  cosS+-^  =  sin2S,  (31) 

a*      ab  IP 


we  see  that  it  represents  an  ellipse  (since  —  •  —  --  —  -  =  [  —  —  ) 

\  ab  J 


is  positive)  whose  centre  is  at  the  origin.     The  resultant  motion 
is  therefore  called  elliptic  harmonic  motion. 


112 


KINEMATICS.  [207. 


207.  Although  in  what  precedes  we  have  assumed  the  axes 
at  right  angles  to  each  other,  this  is  not  essential.     The  same 
equation  (31)  is  obtained  for  oblique  axes  Ox,  Oy,  and  it  is  easy 
to  show  (say  by  transforming  (31)  to  rectangular  axes)  that  this 
equation  still  represents  an  ellipse.     We  have,  therefore,  the 
general  result  that  any  number  of  simple  harmonic  motions  .of 
the  same  period  and  in  the  same  plane,  whatever  may  be  their 
directions,  amplitudes,  and  phases,  compound  into  a  single  elliptic 
harmonic  motion. 

208.  A  few  particular  cases  may  be  noticed.     The  equation 
(31)  will  represent  a  (double)  straight  line,  and  hence  the  elliptic 
vibration   will   degenerate   into   a   simple   harmonic   vibration, 
whenever  sin28  =  o,  i.e.  when  8  =  n7r,  where  n  is  a  positive  or 
negative   integer.     In  this   case  cosS  is   +i  or  —  i,  and  (31) 
reduces  to 

-  —  ^  =  0,    if    S  = 

a     b 


and  to 


Thus  two  rectangular  vibrations  of  the  same  period  compound 
into  a  simple  harmonic  vibration  when  they  differ  in  phase  by 
an  integral  multiple  of  TT,  that  is  when  one  lags  behind  the 
other  by  half  a  wave  length. 

209.  Again,  the  ellipse  (31)  reduces  to  a  circle  only  when 
cosS=o,  i.e.  $=(2n+i)7r/2,  and  in  addition  a  =  b,  the  co-ordi- 
nates being  assumed  orthogonal. 

Thus  two  rectangular  vibrations  of  equal  period  and  ampli- 
tude compound  into  a  circular  vibration  if  they  differ  in  phase 
by  7T/2,  i.e.  if  one  is  retarded  behind  the  other  by  a  quarter  of 
a  wave  length. 

This  circular  harmonic  motion  is  evidently  nothing  but  uni- 
form motion  in  a  circle;  and  we  have  seen  in  Art.  172  that, 
conversely,  uniform  circular  motion  can  be  resolved  into  two 


2ii.]  PLANE   MOTION.  II3 

rectangular  simple  harmonic   vibrations    of   equal   period   and 
amplitude,  but  differing  in  phase  by  7r/2. 

The  results  of  Arts.  205-209  can  also  be  established  by 
purely  geometrical  methods  of  an  elementary  character.* 

210.  It  remains  to  consider  the  case  when  the  given  simple 
harmonic  motions  do  not  all  have  the  same  period.     It  follows 
from  Art.  189  that  in  this  case,  if  we  again  project  the  given 
motions  on  two  rectangular  axes  Ox,  Oy,  the  resulting  motions 
along  Ox,  Oy  are  in  general  not  simply  harmonic. 

The  elimination  of  t  between  the  expressions  for  x  and  y  may 
present  difficulties.  But,  of  course,  the  curve  can  always  be 
traced  by  points,  graphically. 

We  shall  here  consider  only  the  case  when  the  motions  along 
Ox  and  Oy  are  simply  harmonic. 

211.  If  two  simple  harmonic  motions  along  the  rectangular 
directions  Ox,  Oy,  viz. : 


x=a±  cos  — - 


of  different  amplitudes,  phases,  and  periods  are  to  be  com- 
pounded, the  resulting  motion  will  be  confined  within  a  rec- 
tangle whose  sides  are  2ap  2a2,  since  these  are  the  maximum 
values  of  2x  and  2y. 

The  path  of  the  moving  point  will  be  a  closed  curve  only  when 
the  quotient  T^T2  is  a  commensurable  number,  say  =  m/n, 
where  m  is  prime  to  n.  The  x  co-ordinate  of  the  curve  will 
have  m  maxima,  the  y  co-ordinate  n,  and  the  whole  curve  will 
be  traversed  after  m  vibrations  along  Ox  and  n  along  Oy. 

The  formation  of  the  resulting  curve  will  best  be  understood 
from  the  following  example. 


*  See,  for  instance,  J.  G.  MACGREGOR,  An  elementary  treatise  on  kinematics  and 
dynamics,  London,  Macmillan,  1887,  pp.  115  sq. 

PART   I — 8 


II4  KINEMATICS.  [212. 

212.  Let.  al  =  az=af  e1  =  o,  e2  =  &,  and  let  the  ratio  of  the 
periods  be  T1/T2  =  2/i.  The  equations  of  the  component  sim- 
ple harmonic  vibrations  are 

x=acoso)t,  y  —  a  cos(2a)t+S). 
Here  it  is  easy  to  eliminate  t.     We  have 

y  =  a  cos  2  tot  cos  B  —  a  sin  2  a>t  sin  S 


—  ni'y*        i\rn*%      o 

—  a\   ^  ~o         L       ^Ub  °         ^ ifr  ~  \/  x  o 

a*       I  a^        or 


Hence  the  equation  of  the  path  is  : 

ay  =  (23?  —  a2)  cos  S  —  2 x^/cP—x*'  sin  B. 


If  there  be  no  difference  of  phase  between  the  components, 
i.e.  if  S=o,  this  reduces  to  the  equation  of  a  parabola : 


For  S  =  ?r/2,  the  equation  also  assumes  a  simple  form  : 


213.  It  is  instructive  to  trace  the  resulting  curves  for  a  given 
ratio  of  periods  and  for  a  series  of  successive  differences  of 
phase  (Lissajouss  Curves}. 

Thus,  in  Fig.  51,  the  curve  for  7\/  7^  =  3/4,  and  for  a  phase 
difference  S  =  o  is  the  fully  drawn  curve,  while  the  dotted  curve 
represents  the  path  for  the  same  ratio  of  the  periods  when  the 
phase  difference  is  one-twelfth  of  the  smaller  period.  The 
equations  of  the  components  are  for  the  full  curve 


3  4 

and  for  the  dotted  curve 


2I3-] 


PLANE   MOTION. 


In  tracing  these  curves,  imagine  the  simple  harmonic  motions 
replaced  by  the  corresponding  uniform  circular  motions  (Fig. 
51).  With  the  amplitudes  6,  5  as  radii,  describe  the  semi- 
circles ADB,  AEC,  so  that  BC  is  the  rectangle  within  which 
the  curves  are  confined ;  the  intersection  of  the  diagonals  of 
this  rectangle  is  the  origin  O,  AB  is  parallel  to  the  axis  of  xt 
AC  to  the  axis  of  y.  Next  divide  the  circles  over  AB,  AC  into 
parts  corresponding  to  equal  intervals  of  time.  In  the  present 
case,  the  periods  for  AB,  AC  being  as  3  to  4,  the  circle  over 


Fig.  51, 


AB  must  be  divided  into  3/2  equal  parts,  that  over  AC  into 
4 «.  In  the  figure,  n  is  taken  as  4,  the  circles  being  divided 
into  12  and  16  equal  parts,  respectively. 

The  first  point  of  the  full  drawn  curve  corresponds  to  ^=o, 
that  is  x=6,y  =  $  ;  this  gives  the  upper  right  hand  corner  of 
the  rectangle.  The  next  point  is  the  intersection  of  the  vertical 
line. through  D  and  the  horizontal  line  through  E,  the  arcs 
BD=i/i2  of  the  circle  over  AB,  and  CE=i/i6  of  that  over 
AC  being  described  in  the  same  time,  so  that  the  co-ordinates 
of  the  corresponding  point  are 


u6 


KINEMATICS. 


[214. 


—     i/=    cos   27T--; 


[  27T--4;) 
\        i6/ 


Similarly  the  next  point 


is  found  from  the  next  two  points  of  division  on  the  circles,  etc. 
To  construct  the  dotted  curve,  it  is  only  necessary  to  begin 
on  the  circle  over  AB  with  D  as  first  point  of  division. 

214.  Exercises. 

(1)  With  the  data  of  Art.  213  construct  the  curves  for  phase  differ- 
ences of  2/12,  3/i2,  •••  11/12  of  the  smaller  period. 

(2)  Construct  the  curves  (Art.  212) 

x  =  a  cos  to/,  y  =  a  cos(2o>/  +  S) 

for    8=  O,  7T/4,  7T/2,   37T/4,  TT,   511/4,   3W/2,    771/4,    27T. 

(3)  Trace  the  path  of  a  point  subjected  to  two  circular  vibrations  of 
the  same  amplitude,  but  differing  in  period,  (a)  when  the  sense  is  the 
same;  (b)  when  it  is  opposite. 

215.  The  mathematical  pendulum  is  a  point  compelled  to  move 

in   a   vertical   circle   under   the 
acceleration  of  gravity. 

Let  <9  be  the  centre  (Fig.  52), 
A  the  lowest,  and  B  the  highest 
point  of  the  circle.  The  radius 
OA  =  loi  the  circle  is  called  the 
length  of  the  pendulum.  Any 
position  P  of  the  moving  point 
is  determined  by  the  angl< 
AOP  =  6  counted  from  the  ver- 
tical radius  OA  in  the  positiv< 
(counterclockwise)  sense  of  rota- 
tion. 
If  PQ  be  the  initial  position  of  the  moving  point  at  the  time 


2i6.]  PLANE   MOTION. 


/=0,  and  £  AOPQ  =  0Q,  then  the  arc  P0P=s  described  in  the 
time/  is  j  =  /(00-0);  hence  v=ds/dt=  -IdO/dt,  and  dfr/dT/ 
=  —Id26/dt*,  the  negative  sign  indicating  that  0  diminishes  as 
j  and  /  increase. 

Resolving  the  acceleration  of  gravity,  g>  into  its  normal  and 
tangential  components  gcosO,  £-sin#,  and  considering  that  the 
former  is  without  effect  owing  to  the  condition  that  the  point 
is  constrained  to  move  in  a  circle,  we  obtain  the  equation  of 
motion  in  the  form  dv/dt  =g  sin  0,  or 


(32) 


216.  The  first  integration  is  readily  performed  by  multiplying 
the  equation  by  dQjdt  which  makes  the  left-hand  member  an 
exact  derivative,  , 


hence  integrating,  we  obtain 


or  considering  that  v=—ldQldt, 


To  determine  the  constant  C,  the  initial  velocity  ^0  at  the 
time  t=o  must  be  given.     We  then  have  J^02 
hence 


cos  e=g-lcvs  00  +  /cos  0      (33) 


The  right-hand  member  can  readily  be  interpreted  geometri- 
cally ;  v^/2g  is  the  height  by  falling  through  which  the  point 
would  acquire  the  initial  velocity  VQ  (see  Art.  113);  /cos# 


Hg  KINEMATICS.  [217. 

-fcos00=OQ.-OQQ=Q0Q,  if  2,  (?0  are  the  projections  of  P, 
P0  on  the  vertical  AB.  If  we  draw  a  horizontal  line  MN  at 
the  height  v£/2g  above  PQ  and  if  this  line  intersect  the  vertical 
AB  in  R,  we  have  for  the  velocity  v  the  expression  : 


(34) 
If  the  initial  velocity  be  =o,  the  equation  would  be 

(35) 


At  the  points  M,  N  where  the  horizontal  line  MN  inter- 
sects the  circle  the  velocity  becomes  o.  The  point  can  there- 
fore never  rise  above  these  points. 

Now,  according  to  the  value  of  the  initial  velocity  z/0,  the  line 
MN  may  intersect  the  circle  in  two  real  points  M,  N,  or  touch 
it  at  B,  or  not  meet  it  at  all.  In  the  first  case  the  point  P 
performs  oscillations,  passing  from  its  initial  position  PQ  through 
A  up  to  M,  then  falling  back  to  A  and  rising  to  N,  etc.  In  the 
third  case  P  makes  complete  revolutions. 

217.  The  second  integration  of  the  equation  of  motion  cannot 
be  effected  in  finite  terms,  without  introducing  elliptic  func- 
tions. But  for  the  case  of  most  practical  importance,  viz.  for 
very  small  values  of  6,  it  is  easy  to  obtain  an  approximate  solu- 
tion. In  this  case  6  can-  be  substituted  for  sin#,  and  the 
equation  becomes  : 

••":    '  ™+70=°-        '-         <36> 

This  is  a  well  known  differential  equation  (compare  Art.  122, 
Eq.  (19),  and  Art.  176),  whose  general  integral  is 


sn 


6  =  £  cos  / 


218.] 


PLANE   MOTION. 


119 


The  constants  Clt  C2  can  be  determined  from  the  initial  con- 
ditions for  which  we  shall  now  take  0  =  #0  and  v  =  o  when  /=o; 
this  gives  £\  =  00>  C2  =  o  ;  hence 


(37) 


The  last  equation  gives  for  0=  —  00  the  time 
tion,  or  half  the  period  Tt 


of  one  oscilla- 


(38) 


The  time  of  a  small  oscillation  is  thus  seen  to  be  indepen- 
dent of  the  arc  through  which  the  pendulum  swings ;  in  other 
words,  for  all  small  arcs  the  times  of  oscillation  of  the  same 
pendulum  are  the  same ;  such  oscillations  are  therefore  called 
isochronous. 

218.  A  pendulum  whose  length  is  so  adjusted  as  to  make  it 
perform  at  a  certain  place  just  one  oscillation  in  a  second  is 
•called  a  seconds  pendulum. 

Putting  ^=  i  in  (38)  we  find  for  the  length  /x  of  the  seconds 
pendulum  at  a  place  where  the  acceleration  of  gravity  is  g, 


(39) 


As  the  length  of  the  pendulum  can  be  determined  with  great 
accuracy  by  measurement,  the  pendulum  can  be  used  to  find 
the  value  of  g. 

The  length  of  the  seconds  pendulum  is  very  nearly  a  metre  ; 
it  varies  for  points  at  sea  level  from  ^=99.103  cm.  at  the  equa- 
tor to  /1  =  99.6io  at  the  poles.* 


*  Further  numerical  data  for  ^  and  g  will  be  found  in  J.  D.  EVERETT,  C.  G.  S. 
system  of  units,  1891,  pp.  21-22. 


120  KINEMATICS.  [219. 

219.   Let  n  be  the  number  of  small  oscillations  made  by  a 
pendulum  of  length  /in  the  time  7!     Then,  by  (38), 


T  I/ 

~=7r\r 

n          \  sr 


(40) 
g 

If  71  and  one  of  the  three  quantities  n,  /,  g  in  this  equation  be 
regarded  as  constant,  the  small  variations  of  the  two  others  can 
be  found  approximately  by  differentiation.  For  instance,  if  the 
daily  number  of  oscillations  of  a  pendulum  of  constant  length 
be  observed  at  two  different  places,  we  have,  since  T  and  /  are 
constant, 

Tj  TT^Jdg 

--  -dn  =  ---  *, 
n*  2     gl 

or,  dividing  by  (40), 

^  =  1*  (41) 

n       2g 

220.   Exercises. 

(1)  Find  the  number  of  oscillations  made  in  a  second  and  in  a  day 
by  a  pendulum  i  metre  long,  at  a  place  where  £-=981.0. 

(2)  Find  the  length  of  the  seconds   pendulum  at  a  place  where 
#=32.12. 

(3)  To  determine  the  value  of  g  at  a  given  place,  the  length  of  a 
pendulum  was  adjusted  until  it  would  make  86  400  oscillations  in  24 
hours.     Its  length  was  then  found  to  be  3.3031  feet.     What  was  the 
value  of  g  ? 

(4)  A  chandelier  suspended  from  the  ceiling  of  a  theatre  is  seen  to 
vibrate  24  times  a  minute.     Find  its  distance  from  the  ceiling. 

(5)  A  pendulum  adjusted   so  as  to  beat   seconds  at   the  equator 
(^=978.1)  is  carried  to  another  latitude  and  is  there  found  to  make 
100  oscillations  more  per  day  ;  find  the  value  of  g  at  this  place. 

(6)  Investigate  whether  the  approximate  process  of  Formula  (41)  is 
sufficiently  accurate  for  the  solution  of  Ex.  (5). 

(7)  If  the  length  of  a  pendulum  be  increased  by  a  small  amount  dlt 
show  that  the  daily  number  of  oscillations,  «,  will  be  decreased  so  that 


2  / 


221.]  PLANE  MOTION.  I2r 

(8)  A  clock  is  gaining  3  minutes  a  day.     How  much  should  the 
pendulum  bob  be  screwed  up  or  down  ? 

(9)  A  clock  regulated  at  a  place  where  £-=32.19  is  carried  to  a 
place  where  ^-=3  2.  1  4.     How  much  will  it  gain  or  lose  per  day  if  the 
length  of  its  pendulum  be  not  changed  ? 

(10)  The  acceleration  of  gravity  being  inversely  proportional  to  the 
square  of  the  distance  from  the  earth's  centre,  show  that  the  seconds 
pendulum  will  lose  about  22  seconds  per  day  if  taken  to  a  height  one 
mile  above  sea  level. 

(u)    A  seconds  pendulum  loses  12  seconds  per  day,  if  taken  to  the 
top  of  a  mountain.     What  is  the  height  of  the  mountain  ? 

(12)    Show  that  for  small   oscillations  the  motion  of  a  pendulum 
is  nearly  simply  harmonic,  and   deduce   from   this  fact   the   equation 


221.  When  the  oscillations  of  a  pendulum  are  not  so  small 
that  the  arc  can  be  substituted  for  its  sine  as  was  done  in  Art. 
217,  an  expression  for  the  time  t^  of  one  oscillation  can  be 
obtained  as  follows. 

We  have  by  (33),  Art.  216, 

cos0.  (33) 


Let  the  time  be  counted  from  the  instant  when  the  moving 
point  has  its  highest  position  (TV  in  Fig.  52),  so  that  ^0=o. 
Substituting  v—  —IdQ/dt  and  applying  the  formula 

cos  0=i—2  sin2  J  9 
we  find  : 


whence 

d9 


(43) 


122 


KINEMATICS. 


[221. 


Integrating  from  0  =  o  to  0  =  00  and  multiplying  by  2  we  find 
for  the  time  ^  of  one  oscillation  : 

dO 


o       /  . 

\/sm 
\ 


.  2<90 

2  —  — 


0 

sm2- 


(43) 


As  0  cannot  become  greater  than  00  we  may  put  sin 
=  sin  (00/2)  sin  </>,  thus  introducing  a  new  variable  <f>  for  which 
the  limits  are  o  and  ir/2.  Differentiating  the  equation  of  sub- 
stitution, we  have 


6 


— 


or,  as  cos(0/2)=  Vi  —  sin2(00/2)  sin2</>, 
,2  sin—  cos  <f)  d<f> 

de= 


Substituting  these  values  and  putting  for  shortness 


sin  —  =  K, 


(44) 


we  find  for  the  time  ^  of  one  oscillation  : 


ti  =  2\l-    i   '     ,       -r  •  (45)   : 

Vi-*2sin2<£ 

The  integral  in  this  expression  is  called  the  complete  elliptic  I 
integral  of  the  first  species,  and  is  usually  denoted  by  K.     Its 
value  can  be  found  from  tables  of  elliptic  integrals  or  by  ex-  | 
panding  the  argument  into  an  infinite  series  by  the  binomial  I 
theorem  (since  K  sin  <£  is  less  than  i),  and  then  performing  the 
integration.     We  have 

1'3   4   • 

—  K  sin  £      •••> 


hence 


(46) 


223.]  PLANE   MOTION. 


I23 


If  H  be  the  height  of  the  initial  point  N  (0  =  00)  above  the 
lowest  point  A  of  the  circle,  we  have  by  (44) 

0  0A      i  —  cos  0n     H 


-       -     —       —  ,, 

2  2  2/ 

so  that  (46)  can  be  written  in  the  form 

7 


222.  Exercises. 

(  i  )    Show  that  4  =  7r\/-(i  +-i  +  -5-  +     225     .f  ...  )  if  the  ampli- 

VV       l6      I024      147456          / 
tude  200  of  the  oscillation  is  120°. 

(2)  Show  that  as  a  second  approximation  to  the  time  of  a  small 
oscillation  we  have  ti=ir^t/g(i-t-00z/i6). 

(3)  Find  the  time  of  'oscillation  of  a  pendulum  whose  length  is  i 
metre  at  a  place  where  £-=980.8,  to  four  decimal  places. 

(4)  A  pendulum  hanging  at  rest  is  given  an  initial  velocity  v^.     Find 
to  what  height  h\  it  will  rise. 

(5)  Discuss  the  pendulum  problem  in  the  particular  case  when  MN 
(Fig.  52)  touches  the  circle  at  B,  that  is  when  the  initial  velocity  is  due 
to  falling  from  the  highest  point  of  the  circle. 

223.  Central   Motion.     The  motion    of    a   point   P   is    called 
central   if   the   following   two  conditions  are   fulfilled  :  (i)  the 
direction  of   the  acceleration  must    pass  constantly  through  a 
fixed  point  O  ;  (2)  the  magnitude  of  the  acceleration  must  be  a 
function  of  the  distance  OP  =r  only,  say 

/=/«.  (48) 

The  fixed  point  O  is  in  this  case  usually  regarded  as  the  seat 
of  an  attractive  or  repulsive  force  producing  the  acceleration, 
and  is  therefore  called  the  centre  of  force. 

Harmonic  motion  as  discussed  in  Arts.  172-214  is  a  special 
case  of  central  motion,  viz.  the  case  in  which  the  acceleration/  is 


KINEMATICS.  [224. 

directly  proportional  to  the  distance  from  the  fixed  centre  O,  i.e. 
f(r)=»r. 

Another  very  important  particular  case  is  that  of  planetary 
motion  in  which  f(r)=/j,/r2;  this  will  be  discussed  below, 
Arts.  236,  239. 

We  proceed  to  establish  the  fundamental  properties  of  central 
motion. 

224.  The  motion  is  fully  determined  if   in  addition    to   the 
f<>rm  of  the  f  unction /(r)  we  know  the  "initial  conditions,"  say 
the  initial  distance  OPQ  =  rQ  (Fig.  53)  and  the  initial  velocity  v^ 
of  the  point  at  the  time  /=o.     As  ^0  must  be  given  both  in 
magnitude  and  direction,  the  angle  ^0  between  r0  and  VQ  must 
be  known. 

225.  It  is  evident,  geometrically,  that  the  motion  is  confined 
to  the  plane  determined   by  O  and  VQ  since   the   acceleration 


Fig.  53. 

always  lies  in  this  plane.  This  fact  that  the  motion  is  plane 
depends  solely  on  the  former  of  the  two  conditions  of  our 
problem  (Art.  223) ;  that  is,  any  motion  in  which  the  acceleration 
passes  constantly  through  a  fixed  point  is  plane. 

226.  With  O  as  origin,  let  x,  y  be  the  rectangular  Cartesian 
co-ordinates  of  the  moving  point  Pt  and  r,  6  its  polar  co-ordinates, 
at  any  time/.  Then  cos#=;r/r,  sin  0=y/r  are  the  direction 
cosines  of  OP=r,  and,  therefore,  those  of  the  acceleration/, 


228.]  PLANE   MOTION.  12$ 

provided  the  sense  of  j  be  away  from  the  centre,  i.e.  provided 

the  force  causing  the  acceleration  be  repulsive.     In  the  case  of 

attraction^  the  direction  cosines  of  j  are  of  course  —x/r,  —y/r. 

Thus  the  equations  of  motion  are  in  the  case  of  attraction, 


r/  \  / 

~f(r}r  (49) 

For  repulsion,  it  would  only  be  necessary  to  change  the  sign  of 


227.  To  perform  a  first  integration,  multiply  the  equations 
(49)  by  y,  x  and  subtract  when  the  left-hand  member  will  be 
found  an  exact  derivative,  while  the  right-hand  member  van- 
ishes. Hence,  integrating  and  denoting  the  constant  of  integra- 
tion by  h,  we  find 

,     ,f-,f  =,,  ^       (50) 

or,  introducing  polar  co-ordinates, 


These  equations  show  that  the  sectorial  velocity  is  constant, 
and  —\h  for  our  problem  (see  Art.  135  and  Art.  163,  Ex.  (4)). 

228.  Let  5  be  the  sector  P0OP  described  by  the  radius  vector 
r  in  the  time  t,  so  that  dS=^r*d9.  Then  (5  1)  can  be  written  in 
the  form 


whence  integrating 

S=\ht;  (53) 

this  expresses  the  fact  that  the  sector  is  proportional  to  the  time 
in  which  it  is  described  which  is  of  course  only  another  way  of 
stating  the  proposition  of  Art.  227. 


126  KINEMATICS.  [229. 

The  proof  of  the  converse,  viz.  that  if  in  a  plane  motion 
the  areas  swept  out  by  the  radius  vector  drawn  from  a  fixed 
point  be  proportional  to  the  time,  the  acceleration  must  con- 
stantly pass  through  that  fixed  point,  is  left  to  the  student. 

229.  It  is  well  known  that  Kepler  had  found  by  a  careful 
examination  of  the  observations  available  to  him  that  the  orbits 
described  by  the  planets  are  plane  curves,  and  the  sector  described 
by  the  radius  vector  drawn  from  the  sun  to  any  planet  is  propor- 
tional to  tJie  time  in  which  it   is   described.     This   constitutes 
Kepler's  first  law  of  planetary  motion. 

He  concluded  from  it  that  the  acceleration  must  constantly 
pass  through  the  sun. 

230.  To  express  the  value  of  the  constant  of  integration  h  in  ; 
terms  of  the  given  initial  conditions  (Art.  224),  i.e.  by  means 
of  r0,  v§,  i/rQ,  we  notice  that,  at  any  time  /, 

dO         rdQ   ds  /     ^ 

'v'         (54) 


hence  for  the  time  /=o,  we  find 

^  =  Vosin^o-  (55) 

Denoting  the  perpendiculars  let  fall  from  O  on  v§  and  v  by  ; 
/0,  /,  we  have  r0sin^0=/0,  rsin-v/r=/;  hence  also 

k=PM=pv,  (56) 

i.e.  the  velocity  at  any  time  is  inversely  proportional  to  its  distance  \ 
from  the  centre. 

231.  The  equations  of  motions  (49),  if  multiplied  by  dx/dt, 
dy/dt  and  added,  give  an  equation  in  which  both  members  are 
exact  derivatives.  On  the  left  we  find 


233-]  PLANE   MOTION. 

on  the  right 


dt        dt  2r  dr  2r      dt  dt 

The  equation 


(57). 
can  therefore  be  integrated  and  we  obtain 

M*-.     :,       (58) 


232.  The  two  methods  of  integrating  the  differential  equa- 
tions of  motion  used  in  Art.  227  and  in  Art.  231  are  known, 
respectively,   as  the  principle   of  areas  and   the   principle  of 
energy  (or  vis  viva).     The  former  name  explains  itself.     The 
latter  is  due  to  the  fact  (to  be  more  fully  explained  in  kinetics) 
that  if  equation  (58)  be  multiplied  by  the  mass  of  the  moving 
point,  the  left-hand  member  will  represent  the  increase  of  the 
kinetic  energy  of  the  point  during  the  motion. 

Each  of  these  methods  of  preparing  the  equations  of  motion 
for  integration  consists  merely  in  combining  the  equations  so 
as  to  obtain  an  exact  derivative  in  the  left-hand  member  of  the 
resulting  equation.  If  by  this  combination  the  right-hand 
member  happens  to  vanish  or  to  become  likewise  an  exact 
derivative,  an  integration  can  at  once  be  performed.  This  is 
the  case  in  our  problem. 

233.  The  two  equations  (51)  and  (58)  can  be  used  to  find 
the  equation  of  the  path.     We  have  for  any  curvilinear  motion 
(by  (4),  Art.  142) 


eliminating  dt  by  means  of  (51)  this  becomes 


128  KINEMATICS.  [234. 

where  u—\/r.  Substituting  for  v  its  expression  in  terms  of 
r  or  #,  from  (58),  we  have  the  differential  equation  of  the  path 
which  is  directly  integrable. 

Shorter  methods  will  often  suggest  themselves  in  particular 
cases. 

234.  To  solve  the  -converse  problem,  viz.  to  find  the  law  of 
acceleration  when  the  equation  of  the  path  is  given,  we  havei 
only  to  substitute  in  (57)  the  expression  of  ^  from  (59).  We 
find,  with  u—\lrt 


dr  du     dr  du 


235.  Kepler  in  his  second  law  had  established  the  empirical 
fact  that  the  orbits  of  the  planets  are  ellipses,  with  the  sun  at 
one  of  the  foci. 

From  this,  Newton  concluded  that  the  law  of  acceleration 
must  be  that  of  the  inverse  square  of  the  distance  from  the  sun. 

Equation  (60)  allows  us  to  draw  this  conclusion.  The  polar 
equation  of  an  ellipse  referred  to  focus  and  major  axis  is 


where  t=d>2/a=a(i—e2);   a,  b  being  the  semi-axes,  /the  semi- 
latus  rectum,  and  e  the  eccentricity  of  the  ellipse.     Hence 


.and  (60)  becomes 

236.  The  third  law  of  Kepler,  found  by  him  likewise  as  an 
empirical  fact,  asserts  that  the  squares  of  the  periodic  times  of 
-different  planets  are  as  the  cubes  of  the  major  axes  of  their  orbits. 


238.] 


PLANE   MOTION. 


I29 


From  this  fact  Newton  drew  the  conclusion  that  in  the  law 
of  acceleration, 

I  J=fV)=%         '    .  •  :  (62) 

1  the  constant  /UL  has  the  same  value  for  all  the  planets. 

Our  formulas  show  this  as  follows.  Let  T  be  the  periodic 
time  of  any  planet,  i.e.  the  time  of  describing  an  ellipse  whose 
.semi-axes  are  a,  b.  Then,  since  the  sector  described  in  the 
time  T  is  the  area  irab  of  the  whole  ellipse,  we  have  by  (53) 


Substituting  in  (61)  the  value  of  h  found  from  this  equation 
we  have 


Hence 


is  constant  by  Kepler's  third  law. 


(63) 
(64) 


237.   As  mentioned  in  Art.  230,  the  velocity  v  can  be  ex- 
pressed in  terms  of  the  perpendicular/  let  fall  from  the  centre 
on  the  tangent  to  the  path : 


= 


(65) 


The  acceleration  /  is  also  conveniently  expressible  in  terms 
•of/.     We  have  by  (57) 


dr 


=  _i  ffi*.(L\  =  # 

2  3 


dr 


(66) 


238.  Finally,  another  expression  for  the  acceleration  is  some- 
times found  convenient.  In  any  motion,  the  component  of  the 
acceleration  along  the  radius  vector  is  (see  Art.  161) 


J 

PART  I — 9 


.      <Pr       (dOV 

j    •— A*I   I 

dt*       \dt) 


KINEMATICS.  [239. 

As  in  our  problem  the  total  acceleration  is  along   the   radius 
vector,  in  the  sense  towards  the  origin,  we  have 


or  since,  by  (51), 


The  first  term  is  what  the  acceleration  would  be  if  the  motion 
were  rectilinear  along  the  radius  vector  ;  the  second  term 
represents  what  is  due  to  the  curvature  of  the  path. 

239.  Planetary  Motion,  in  its  simplest  form,  is  (see  Art.  223) 
that  particular  case  of  central  motion  in  which  the  acceleration 
is  inversely  proportional  to  the  square  of  the  distance  from  the 
centre  O,  so  that 


where  //,  is  a  constant,  viz.  the  acceleration  at  the  distance  r=  I 
from  O. 

The  equations  of  motion  (49)  are  in  this  case,   with  O  as 
origin, 

d^x  _  _     x_     d^y  _  _    y_  /^gx 

Combining  these  by  the  principle  of  energy  (Arts.  231,  232), 
we  find 


dt  rs\  dt     ' dt)         r*  dt\     2 

<® 


_ 

r*dt  dt    ' 

hence  integrating 


£-IL  (69) 

T       rn 


24I-]  PLANE   MOTION.  !3I 

240.   To  find  the  equation  of  the  path,  or  orbit,  we  write  the 
equations  (68)  in  the  form 


and  eliminate  r2  by  means  of  (51): 


dt*         h          dt     dP         h          dt 
These  equations  can  be  integrated  separately 


where  vly  ^2  are  the  components  of  the  initial  velocity. 

Multiplying  by  7,  x  and  subtracting,  we  find,  owing  to  (50), 

(70 

241.  The  geometrical  meaning  of  this  equation  is  that  the 
radius  vector  r=  V-r2+/2  drawn  from  the  fixed  point  O  to  the 
moving  point  P  is  proportional  to  the  distance  of  P  from 
the  fixed  straight  line 

(72) 

It  represents,  therefore,  a  conic  section  having  O  for  a  focus 
and  the  line  (72)  for  the  corresponding  directrix. 

The  character  of  the  conic  depends  on  the  absolute  value  of 
the  ratio  of  the  radius  vector  to  the  distance  from  the  directrix ; 
according  as  this  ratio 


the  conic  will  be  an  ellipse,  a  parabola,  or  a  hyperbola.  The 
criterion  can  be  simplified.  Multiplying  by  p/h  and  squaring, 
we  have 


KINEMATICS.  [242. 

or  since  v    +  v=v     and  //  =  rz>  sinir  =  rz/  : 


242.    Introducing  polar  co-ordinates  in  (71),  the  equation  of 
the  orbit  assumes  the  form 


or  putting  (7^2  —  p)//t2  =  C  cos  a,  vjh  =  C  sin  «, 

.  (74) 


This  equation  might  have  been  obtained  directly  by  integrat- 
ing (60),  which  in  our  case,  with  /(r)=/i/r2,  reduces  to 


the  general  integral  of  this  differential  equation  is  of  the  form 
(74),  C  and  a  being  the  constants  of  integration. 

Equation  (74)  represents  a  conic  section  referred  to  the  focus 
as  origin  and  a  line  making  an  angle  «  with  the  focal  axis  as 
polar  axis. 

243.   Exercises. 

(1)  If  2  k  be  the  chord  intercepted  by  the  osculating  circle  on  the 
radius  vector  drawn  from  the  fixed  centre,  show  that  z?=  k-f(r). 

(2)  A  point  moves  in  a  circle;  if  the  acceleration  be  constant  in 
direction,  what  is  its  magnitude  ? 

(3)  A  point  moves  in  a  circle  ;    if  the  acceleration  be  constantly 
directed  towards  the  centre,  what  is  its  magnitude  ? 

(4)  A  point  is  subject  to  a  central  acceleration  proportional  to  the: 
distance  from  the  centre  and  directed  away  from  the  centre  ;  find  the 
equation  of  the  path. 

(5)  A  point  P  is  subject  to  two  accelerations,  —p?-O±P  directed 
towards  the  fixed  point  O^  and  —p?-O2P  directed  away  from  the  fixed 
point  <92.     Show  that  its  path  is  parabolic. 


245-]  PLANE   MOTION.  133 

(6)  A  point  P  describes  an  ellipse  owing  to  a  central  acceleration 
f(f)  =i*>/r2  directed  toward  the  focus  S.  Its  initial  velocity  VQ  makes  an 
angle  i//0  with  the  initial  radius  vector  r0.  Determine  the  semi-axes  a,  b 
of  the  ellipse  in  magnitude  and  position. 

244.  The  student  will  find  numerous  examples  for  further  practice 
in  the  kinematics  of  a  particle  in  the  following  works  :  P.  G.  TAIT  and 
W.  J.  STEELE,  A  treatise  on  dynamics  of  a  particle,  6th  ed.,  London, 
Macmillan,  1889 ;  W.  H.  BESANT,  A  treatise  on  dynamics,  London,  Bell, 
1885  ;  B.  WILLIAMSON  and  F.  A.  TARLETON,  An  elementary  treatise  on 
dynamics,  2d  ed.,  London,  Longmans,  1889;  W.  WALTON,  Collection  oj 
problems  in  illustration  of  the  principles  of  theoretical  mechanics,  3d  ed., 
Cambridge,  Deighton,  1876. 


5.     VELOCITIES    IN    THE    RIGID    BODY. 

245.  A  rigid  body  is1  said  to  have  plane  motion  when  all  its 
points  move  in  parallel  planes.  Its  motion  is  then  fully  deter- 
mined by  the  motion  of  any  plane  section  of  the  body  in  its 
plane. 

It  has  been  shown  in  Arts.  18-24  that  the  continuous  motion 
of  an  invariable  plane  figure  in  its  plane  consists  in  a  series  of 
infinitesimal  rotations  about  the  successive  instantaneous  cen- 
tres, i.e.  about  the  points  of  the  space  centrode. 

If  at  any  instant  the  centre  of  rotation  and  the  angular  veloc- 
ity to  about  it  be  known,  we  can  find  the  velocity  of  any  point  of 
the  plane  figure. 

To  show  this  let  us  first  take  the  instantaneous  centre  as 
origin.  Then  the  component  velocities  vx,  vy  of  any  point  P 
whose  co-ordinates  are  x,  y,  or  r,  6,  are  found  (Art.  141)  by  dif- 
ferentiating the  expressions 


with    respect   to    /.     Considering    that    dQ/dt   is    the    angular 
velocity  o>  about  the  instantaneous  centre,  we  find 


134 


KINEMATICS. 


[246. 


at 


at 


dt 


dt 


=     cox. 


246.    Next,  taking  an  arbitrary  origin  O  (Fig.  54),  let  x,  y  be 
the  co-ordinates  of  P  and  xl  ',  y'   those  of  any  other  point  Or  of 

the  moving  figure  with  re- 
spect to  fixed  rectangular 
axes  through  O  ;  and  let 
rj  be  the  co-ordinates  of  P 
with  reference  to  rectangu- 
lar axes  through  O'  fixed  in 
the  figure  but  moving  with 
it.  Then,  if  0  be  the  angle 
between  the  axes  Ox  and 
O'g,  we  have 


X' 


-  54- 


x—x]  r  +  f  cos  0—  77  sin0,    y=y' 


sn 


cos#. 


Differentiating  we  find  for  the  component  velocities  of 
parallel  to  the  fixed  axes  Ox,  Oy  : 


dx' 


dO 


Now,  d6/dt  is  the  angular  velocity  o>  about  the  point  O'  while 
dx' /dt,  dy' /dt  are  the  velocities  of  O'  parallel  to  the  fixed  axes, 
say  v*,  vy-.  Considering  moreover  that  f  sin  6  +  77  cos  6  =y—y', 
f  cos  6— ?;  sin  0=x— x',  we  have 


(3) 


velocity  of  P  consists,  therefore,  <?/"  /ze/^  parts,  a  velocity  of 
translation  equal  to  that  of  O'  and  a  velocity  of  rotation  equal  to 
t.'i.itof?  about  O'. 


248.] 


PLANE   MOTION. 


135 


247.  The  instantaneous  centre  being  the  point  whose  velocity 
is  zero  at  the  given  instant,  we  find  its  co-ordinates  XQ,  y^  from 
the  equations 

o  =  vx,  -  ( j/0  -/)  o>,    o  =  zv + (*b  -*')  co, 


whence 


(4) 


By  eliminating  t  between  these  equations,  the  equation  of  the 
space  centrode  can  be  found. 

The  co-ordinates  f0,  ?70  of  the  instantaneous  centre  referred  to 
the  moving  axes  are  found  in  a  similar  way  from  the  equations  (2)  : 


cos 


cos  e  +  v,  sin  0),        (5) 


from  which  the  body  centrode  can  be  found  by  eliminating  t. 

248.  In  Arts.  245  and  246  expressions  were  found  for  the 
component  velocities  vx,  vy  parallel  to  the  fixed  axes  Ox,  Oy. 
To  find  the  component  velocities 
v&  v^  parallel  to  the  moving  axes 
<9f,  Or],  let  x,  y  be  the  co-ordi- 
nates of  any  point  P  with  respect 
to  the  fixed  axes  (Fig.  55),  £,  rj 
those  with  respect  to  the  moving 
axes,  and  let  6  be  the  angle  xO%. 
The  velocity  of  P  parallel  to  the 
axes  O%,  Or)  consists  of  two  parts, 
that  arising  from  the  motion  of  P  relative  to  f  Orj  whose  com- 
ponents are  of  course  d%/dt,  dy/dt,  and  that  due  to  the  rotation 
of  the  moving  axes.  The  components  of  the  latter  velocity  are, 
by  (i),  Art.  245,  —cor),  cog.  Hence 


Fig.  55. 


dt 


dt 


(6) 


KINEMATICS.  [249. 


249.  Exercises. 


(1)  Two  points  A,  A1  of  a  plane  figure  move  on  two  fixed  circles 
described  with  radii  a,  a'   about  O,  O'  ;  show  that  the  angular  velocities 
w,  <i>'  of  OA,  O'A'  about  O,  O'  are  inversely  proportional  to  OM,  O'M, 
M  being  the  point  of  intersection  of  OO'  with  A  A1. 

(2)  Given  the  magnitudes  v,  v'  of  the  velocities  of  two  points  A,  A( 
of  an  invariable  plane  figure  and   the   angle  (v,  v')  formed   by  their 
directions  ;  find  the  instantaneous  centre   C  and  the   angular  velocity 
CD  about  C. 

(3)  Show  that  in   the  "  elliptic  motion  "  of  a  plane  figure    (Arts. 
25-27)  the  velocity  of  any  point  (x',  y')  is 


v  =  [a2  +  x12  +/2-  2a(x'  cos  2  <j>  +/  sin  2 

dt 

(4)    In  the  same  motion  find  the  velocities   of  B  and  O'  (Fig.  6, 
Art.  26)  when  ^4  moves  uniformly  along  the  axis  of  x. 


250.  The  continuous  motion  of  a  rigid  body  is  called  a  trans- 
lation when  the  velocities  of  all  its  points  are  equal  and  parallel 
at  every  moment  (Art.  9).     All  points  describe  therefore  equal 
and  similar  curves,    and  every  line  of  the  body  remains  par- 
allel to  itself.     The  velocity  v  =  ds/dt  of  any  point  is  called  the 
velocity  of  translation  of  the  body. 

251.  A  rigid  body  can  be  imagined  to  be  subjected  to  several 
velocities  of  translation  simultaneously  ;  the  resulting  motion  is 
a  translation  whose  velocity  is   found  by  geometrically  adding 
the  component  velocities. 

Conversely,  the  velocity  of  translation  of  a  rigid  body  can  be 
resolved  into  components  in  given  directions. 

252.  The  continuous  motion  of  a  rigid  body  is  called  rotation 
when  two  points  of  the  body  are  fixed  ;  the  line  joining  these 
points  is  the  axis  of  rotation.     All  points  excepting  those  on  the' 
axis  describe  arcs  of  circles  whose  centres  lie  on  the  axis. 

The  velocity  of  any  point  P  of  the  body  at  the  distance 
OP—r  from   the   axis    is    v  =  a>r=r  dO/dt,  if  w  =  d0/dt   is   the 


254-]  PLANE   MOTION.  l^ 

angular  velocity  of  the  rigid  body.  The  velocities  of  the  differ- 
ent points  of  the  body  at  any  given  moment  are  therefore 
directly  proportional  to  their  distances  from  the  axis,  and  the 
velocities  of  all  points  at  this  moment  are  known  if  the  instan- 
taneous angular  velocity  co  is  given.  It  is  frequently  convenient 
to  imagine  this  angular  velocity  represented  by  its  rotor,  i.e.  by 
a  length  co  laid  off  in  the  proper  sense  on  the  axis  of  rotation 
(see  Arts.  68,  69). 

253.  The   body   may   have   several    simultaneous   rotations. 
Imagine,  for  instance,  a  top  spinning  about  its  axis  placed  on  a, 
table  or  disc  which  is  made  to  rotate  about  an  axis.     The  result- 
ing motion  can  be  found  by  compounding   the  rotors  in  the 
same  way  in  which  the  rotors  representing  infinitesimal  rotary 
displacements  are  compounded  (Arts.  62,  66,  67);    indeed,  the 
rotor  w  =  d6/dt  of  an  .angular  velocity  is  merely  the  rotor  dd 
divided   by  dt,  and   therefore  identical  with   the  rotor  of  the 
angular  displacement  dQ. 

254.  As  we  are  at  present  concerned  with  plane  motion,  we 
require  only  the  rule  for  the  composition  of  angular  velocities 
about  parallel  axes. 

Dividing  the  equations  (i'")  and  {2'")  of  Art.  66  by  dt,  and 
putting  dO/dt=o>,  ddl/dt=col,  d62/dt=a)2,  we  obtain : 

L-\L  LLn          L-iLn  /_\ 

o)  =  ft)1  +  ft)2,        —  =  — -=— L-^-  (7) 

ft)2  C01  ft) 

Thus,  the  resultant  of  two  angular  velocities  co^  w2  about 
parallel  axes  \,  12  is  an  angular  velocity  co  equal  to  their  algebraic 
sum,  ft)  =  ft)1  +  ft)2,  about  a  parallel  axis\  that  divides  the  distance 
between  \,  12  in  the  inverse  ratio  of  co1  and  &)2. 

Conversely,  an  angular  velocity  co  about  an  axis  /  can  always 
be  replaced  by  two  angular  velocities  colf  co2  whose  sum  is  equal 
to  w  and  whose  axes  llt  /2  are  parallel  to  /  and  so  selected  that 
/  divides  the  distance  between  /lf  /2  inversely  as  o^  is  to  co2. 


KINEMATICS. 


[255. 


56> 


255.  It   may  be   well   to   prove   this    important   proposition 
independently.      Any  point  P  (Fig.    56)  in  a  plane  at  right 

angles  to  the  axes  receives  from 
co1  a  linear  velocity  u>^\  per- 
pendicular to  L^P,  and  from 
co2  a  linear  velocity  «2r2  per- 
~  pendicular  to  L2P,  if  LlP  =  rl, 
L^P  =  r^  These  linear  veloci- 
ties fall  into  the  same  straight  line  only  for  points  situated  on 
the  line  L^L^.  A  point  L  whose  linear  velocity  is  zero,  must 
therefore  lie  on  L1L2  so  that  L1L+LL2  =  L1L2',  moreover,  it 
must  satisfy  the  condition  LlL'Col  =  LL2'co2.  This  gives  the 
above  equations  (7). 

256.  The  resulting  axis  lies  between  Ll  and  L2  when  the 
-components  col}  a>2  have  the  same   sense  ;   when  they  are  of 
opposite  sense,  it  lies  without,  on  the  side  of  the  greater  one 
of  these  components. 

If  G>J  and  o>2  are  equal  and  opposite,  say  co1  —  co,  o>2=  —  co,  the 
resulting  axis  lies  at  infinity  (Art.  67).  Two  such  equal  and 
opposite  angular  velocities  about  parallel  axes  are  said  to  form 
a  rotor-couple  ;  its  effect  on  the  rigid  body  is  that  of  a  velocity 
of  translation  v=LlL2'd6/dt=p-a)  at  right  angles  to  the  plane 
of  the  axes.  The  distance  of  the  rotor,  LlL(i=p,  is  called  the 
arm  of  the  couple,  and  the  product  pco  =  v  its  moment. 

257.  A  velocity  of   translation  v  can    therefore   always   be 
replaced   by   a   rotor-couple  pco  =  v,  whose   axes  have  the  dis-| 
tance  p  and  lie  in  a  plane  at  right  angles  to  v. 

Again,  an  angular  velocity  co  about  an  axis  /  can  be  replaced 
by  an  equal  angular  velocity  co  about  a  parallel  axis  /'  at  the 
distance/  from  /,  in  combination  with  a  velocity  of  translation 
v  =  cop  at  right  angles  to  the  plane  determined  by  /  and  /'. 

It  easily  follows  from  these  propositions  that  the  resultant  of 
-any  number  of  velocities  of  translation,  v,  v',  .  .  .,  parallel  to  the 
same  plane,  and  any  number  of  angular  velocities  co,  co',...,  aboui 


259-]  PLANE   MOTION.  l^ 

axes  perpendicular  to  this  plane  is  always  a  single  angular  veloc- 
ity about  an  axis  perpendicular  to  the  plane  or  a  single  velocity 
of  translation  parallel  to  the  plane. 

6.     APPLICATIONS. 

258.  Kinematics  of  Machinery.     A  large  majority  of  the  cases 
of  motion  that  are  of  importance  in  mechanical  engineering  can 
be  reduced  to  plane  motion. 

At  first  glance  the  application  of  theoretical  kinematics  to 
machines  might  seem  to  lead  to  rather  complicated  problems 
owing  to  the  fact  that  a  machine  is  never  formed  by  a  single 
rigid  body,  but  always  consists  of  an  assemblage  of  several 
bodies  some  of  which  may  even  be  not  rigid  (belting,  springs, 
water,  steam).  The  problem  is,  however,  very  much  simplified 
by  a  characteristic  of  all  machines,  properly  so  called,  that  was 
first  pointed  out  and  insisted  upon  by  recent  writers  on  applied 
kinematics,  in  particular  by  Reuleaux.*  This  characteristic  is 
the  constrainment  of  the  motions  of  the  parts  of  a  machine. 

Thus  Professor  Kennedy  defines  a  machine  as  "a  combination 
of  resistant  bodies  whose  relative  motions  are  completely  con- 
strained, and  by  means  of  which  the  natural  energies  at  our 
disposal  may  be  transformed  into  any  special  form  of  work." 

With  the  latter  clause  of  this  definition  we  are  not  at  present 
concerned ;  it  will  be  considered  in  kinetics.  To  explain  the 
former  in  detail  would  lead  us  too  far  into  the  domain  of  applied 
mechanics.  A  brief  indication  of  the  fundamental  ideas  must 
be  sufficient. 

259.  By  considering  machines  of  various  types  it   appears 
that  the  bodies,  or  elements,  composing  a  machine  always  occur 

*F.  REULEAUX,  Theoretische  Kinematik,  Berlin,  1875;  translated  into  English 
and  edited  by  ALEX.  B.  W.  KENNEDY  under  the  title  Kinematics  of  machinery, 
London,  Macmillan,  1876.  Compare  also  R.  WILLIS,  Principles  of  mechanism, 
London,  Longmans,  2d  ed.  1870  (ist  ed.  1841);  F.  GRASHOF,  Theoretische  Ma- 
schinenlehre,  Vol.  II.,  Leipzig,  Voss,  1883;  L.  BURMESTER,  Lehrbitch  der  Kinematik, 
Leipzig,  Felix,  1888;  ALEX.  B.  W.  KENNEDY,  Mechanics,  of  machinery,  London, 
Macmillan,  1886;  J.  H.  COTTERILL,  Applied  mechanics,  London,  Macmillan,  1884. 


I40  KINEMATICS.  [260. 

in  pairs.  Thus  a  single  rigid  bar  will  form  a  lever  only  when 
taken  in  connection  with  a  support,  or  fulcrum  ;  a  shaft  to  be 
used  in  a  machine  must  rest  in  bearings ;  a  screw  must  turn  in 
a  nut.  To  take  a  more  complex  illustration,  consider  the 
mechanism  formed  by  the  crank  and  connecting  rod  of  a  steam- 
engine  (Fig.  57).  It  may  be  regarded  as  composed  of  three 
pairs,  two  so-called  turning  pairs  at  O  and  A,  and  a  sliding  pair 
at  B  ;  and  these  three  pairs  are  connected  by  three  rigid  bars, 
called  links,  OA,  AB,  OB,  the  last  of  which  is  fixed. 


Fig.  57. 

260.  A  sliding  pair  is  formed  by  two  bodies  so  connected  that 
one  is  constrained  to  have  a  motion  of  translation  relatively  to 
the  other.     A  pin  moving  in  a  groove  or  slot,  a  sleeve  sliding 
along  a  shaft,  are  familiar  examples. 

A  turning  pair  constrains  one  body  to  rotate  about  a  fixed 
axis  in  another,  as  in  the  case  of  a  shaft  turning  in  its  bearings. 

A  twisting  pair  makes  one  body  have  a  screw  motion  about 
an  axis  fixed  in  the  other. 

These  three  pairs  are  the  only  so-called  lower  pairs.  They 
are  characterized  as  such  by  the  fact  that  their  elements  have 
surface  contact,  and  that,  if  either  element  be  fixed,  every  point 
of  the  other  is  constrained  to  move  in  a  definite  line.  In  other 
words,  the  constraint  effected  by  lower  pairing  is  such  as  to 
leave  but  one  degree  of  freedom  (see  Art.  37)  to  either  element 
if  the  other  be  fixed. 

261.  All  other  pairs  are  called  higher  pairs.     The  contact  in 
such  pairs  is  usually  line  contact,  and  the  two  bodies  have  more 
than  one  degree  of  freedom  relatively  to  each  other,  usually  two 
degrees,  so  that  if  one  element  be  fixed,  any  point  of  the  other 
is  constrained  to  a  surface. 

Higher  pairs  are  of  far  less  frequent  occurrence  in  ordinary 


263.]  PLANE    MOTION.  !4I 

machines  than  lower  pairs.     The  only  very  common  example  of 
higher  pairing  is  found  in  toothed  wheel  gearing. 

262.  For   the   purposes  of   kinematics   a   machine    may    be 
regarded  as  consisting  of  a  number  of  bodies  (links}  connected 
by  pairs  in  such  a  way  that  when  one  of  the  links  is  fixed  all 
other  links  are    constrained    in  their   motion.     In  most  cases 
this  constraint  is  such  as  to  leave  but  one  degree  of  freedom 
to  every  link. 

A  system  of  links  of  this  kind  forming,  so  to  speak,  a  skeleton 
of  the  machine  is  called  a  kinematic  chain  (Reuleaux).      When 
one  link  of  such  a  chain  is  fixed,  the 
chain   becomes    a   mechanism.     As    a 
typical    example    we    may    take    the 
"  slider  crank  "  in  Fig.  57. 

If  the  pairs  are  all  turning  pairs 
with  parallel  axes,  the  chain  is  called  a 
linkage  (Sylvester).  A  typical  example 
is  the  four  bar  linkage  in  Fig.  58.  A 
linkage  with  one  link  fixed  has  been  called  a  linkwork  (Sylves- 
ter). The  four  bar  linkwork  in  Fig.  58  is  also  called  a  "lever 
crank  "  (Kennedy). 

263.  The  Four  Bar  Linkage  1234  (Fig.  59).    Whatever  may  be 
its  motion,  each  link  considered  separately  moves  as  an  invariable 
plane  figure  and  has  therefore  at  any  moment  an  instantaneous 
centre  C  and  an  angular  velocity  a>  about  this  centre. 

The  centre  £\2  of  I  2  and  the  centre  C2S  of  2  3  must  always  lie 
on  a  line  passing  through  2  since  the  velocity  of  2  is  perpen- 
dicular to  both  6\22  and  ^232. 

Similarly,  3  must  lie  on  the  line  joining  the  centres  C23  and 
CM  ;  and  so  on. 

The  quadrilateral  1234  is  therefore,  and  always  remains, 
inscribed  in  the  quadrangle  CIZCZBCMC^.  This  can  ^e  shown 
to  hold  even  for  the  complete  quadrilateral  and  quadrangle. 
The  complete  quadrilateral,  or  four-side,  1234  has  six  vertices, 
viz.  the  six  intersections  i,  2,  3,  4,  5,  6  of  its  four  sides ;  the 


I42  KINEMATICS.  [263. 

complete  quadrangle,  or  four-point,  C^f^C^C^  nas  s'lx  sides, 
viz.  the  six  lines  C41Clzt  C\>£y&  ^23^34'  ^34^41'  ^12^34'  ^23^41 
joining  its  four  vertices  ;  and  these  six  sides  of  the  quadrangle 
pass  through  the  six  vertices  of  the  quadrilateral,  respectively. 


It  remains  to  prove  that  C1ZC8±  passes  through  5  and  that  C^C^ 
passes  through  6. 

Now  the  velocity  of  2  can  be  expressed  by  Wj  •  C122  and  also 
bya}2-C^2\  hence  Cl22/C2B2  =  a)2/(01;  similarly  C^/C^  = 
ft>3/ft>2.  We  have  therefore,  by  the  proposition  of  Menelaus,*  for 
the  intersection  of  2  3  with  C12CM  : 

5C\2  =  ft)g 

5  Q4     wi 

The  same  value  is  obtained  by  determining  the  intersection  of 
i  4  with  C12C34: ;  the  two  intersections  must  therefore  coincide. 
The  proof  for  the  point  6  is  analogous. 

*  If  the  sides  of  a  triangle  ABC  be  cut  by  any  transversal,  in  the  points  A',  B',  C', 

p>  Al       /""/?'       A  C1 

then  =~-  •  —  •  ~~-  =-i.     See  J.  CASEY,  Sequel  to  Euclid,  London,  Longmans, 
A  C      Jj  A      C  JD 

1882,  p.  69. 


264.]  PLANE   MOTION. 

A  corresponding  proposition  holds  of  course  for  four  bar 
linkages  with  crossed  bars  I  2  4  3,  or  I  3  2  4. 

264.  Lever-crank.  The  linkage  considered  in  the  preceding 
article  becomes  a  mechanism,  or  linkwork,  as  soon  as  one  of  its 
four  links  is  fixed.  It  occurs  in  machines  under  a  variety  of 
forms  some  of  which  are  referred  to  below. 

Let  the  link  3  4  be  fixed ;  then  the  centre  £734  (Fig.  59)  dis- 
appears ;  C^  falls  into  4,  C23  into  3,  and  C12  becomes  the  inter- 
section 5  of  4  i  and  32.  If  i  2  were  fixed  instead  of  34,  34 
would  have  its  centre  at  5. 

Similarly,  if  either  41  or  2  3  be  fixed,  the  centre  of  the  other 
is  6. 


Fig.  60. 

Hence  whichever  of  the  four  links  be  fixed,  the  centres  of 
all  the  links  lie  at  some  of  the  six  vertices  of  the  complete 
quadrilateral  1234. 

If  34  be  the  fixed  link  (Fig.  60),  the  ratio  of  the  angular 
velocities  co1  of  4  i  and  o>2  of  3  2  can  be  found.  For  if  o>  denote 
the  angular  velocity  of  i  2  about  5,  we  have 

4  i  •o)1=5  i -o),     3  2-6>2=5  2-00; 
hence 


*>i     S  2     51     32/41 
or,  by  the  proposition  of  Menelaus : 

^  =  3? 


144  KINEMATICS.  [265. 

265.  Parallelogram:  4  i  =3  2  =  a,  43  =  i  2  =  b  (Fig.  61).  The 
link  i  2  has  evidently  a  motion  of  translation,  its  instantaneous 
centre  lying  at  the  intersection  of  the  parallel  lines  41,  32. 

The  space  centrode  is  the  line  at  infinity ;  the  body  centrode 
may  be  regarded  as  a  circle  of  infinite  radius  described  about 
the  midpoint  of  3  4  as  centre. 


Fig.  61. 

To  find  the  equation  of  the  path  of  any  point  P  rigidly  con- 
nected with  i  2,  let  x,  y  be  the  rectangular  co-ordinates,  with 
respect  to  4  as  origin  and  4  3  as  axis  of  x>  and  x^  y^  its  co-ordi- 
nates for  parallel  axes  through  i  ;  then,  putting  ^  3  4  i  =  6,  we 
have 


hence,  eliminating  0, 


which  represents  a  circle  of  radius  a  whose  centre  has  the  fixed 
co-ordinates  x^  yv 

For  the  velocity  of  P  we  have  dx/dt=  —  aw  sin0,  dyjdt 
=  aa)cos0;  hence  v  =  aa>,  as  is  otherwise  apparent. 

266.  If  in  the  parallelogram  1234  the  point  4  alone  be  fixed, 
we  have  a  linkage  called  the  pantograph. 

It  can  serve  to  trace  a  curve  similar  to  a  given  curve. 
Indeed,  any  line  through  4  (Fig.  62)  cuts  the  opposite  links 


26;.] 


PLANE   MOTION. 


12,  23  (produced  if  necessary)  in  points  A,  A'  whose  paths 
are  homothetic  (similar  and  similarly  situated)  curves.      For 
the  points  4,  A,  A'  remain  always  in 
line  and   the   ratio  4 A/4.A f  remains 
constant.     Hence  if  a  pencil  be  at- 
tached to  Af  and  A  be  made  to  trace 
a  given  curve,  A'  will  trace  a  similar 
curve. 

Instead  of  fixing  4,  the  point  Ar 
might  be  fixed;   then  4  and  A  will 
describe  similar  curves.     This  property  is  utilised  in  Watt's 
parallel  motion  (see  Art.  271). 

The  parallelogram  linkage  furnishes  also  a  simple  instrument 
for  describing  ellipses.  Let  the  sides  of  the  parallelogram  be 
23=41=  #,  12=34=^;  and  let  a  point  A '  on  2  3  produced, 
at  the  distance  b  from  2,  be  fixed  (Fig.  63).  Then,  if  i  be  made 
to  describe  a  straight  line,  passing  through  Af,  4  will  describe 
an  ellipse.  For,  taking  A'  as  origin  and  Ari  as  axis  of  x,  we 


Fig.  62. 


A' 


Fig.  63. 

have  for  the  co-ordinates   of  4:  x=(a-\-2b)  cos0,  y=asm<j>, 
whence 


a2 


267.  In  the  parallelogram  1234,  let  the  link  i  2  be  turned 
so  as  to  coincide  in  direction  with  4  3,  and  then  give  the  links 
4  i  and  3  2  rotations  of  opposite  sense.  We  thus  obtain  a  link- 

PART   I — IO 


I46  KINEMATICS.  [268. 

age  with  equal,  but  intersecting,  opposite  sides,  which  we  may 
call  anti-parallelogram  (Fig.  64).  If  3  4  be  fixed,  the  instanta- 
neous centre  of  i  2  is  the  intersection  5  of  4  i  and  2  3. 


Fig.  64. 

To  obtain  the  centrodes  in  this  case,  notice  that  as  the  tri- 
angles 152  and  534  are  equal,  the  triangle  5  4  2  is  isosceles  ; 
hence  51  =  53,  and  45  —  35=41=0.  The  difference  of  the 
radii  vectores  of  5  drawn  from  4  and  3  being  thus  constant,  it 
follows  that  the  space  centrode  is  a  hyperbola  whose  foci  are 
4,  3,  and  whose  real  axis  =a.  As  43  =  12  =  ^,  the  equation  of 
this  hyperbola  is 


for  4  3  as  axis  of  x  and  the  midpoint  of  4  3  as  origin. 

It  is  easy  to  see  that  the  space  centrode  becomes  an  ellipse 
when  b  <  a. 

As  the  triangles  152  and  354  are  equal  the  body  centrode  is 
an  equal  hyperbola  or  ellipse.  The  two  centrodes  lie  symmet- 
rically with  respect  to  their  common  tangent  at  5. 

For  a  given  anti-parallelogram  the  centrodes  are  hyperbolas 
when  one  of  the  larger  links  is  fixed  ;  they  are  ellipses  when 
one  of  the  shorter  links  is  fixed. 

268.  If  in  the  anti-parallelogram  only  one  point,  say  4,  be 
fixed,  it  can  be  used  as  an  inversor,  i.e.  as  an  instrument  for 
describing  the  inverse  of  a  given  curve 


268.] 


PLANE   MOTION. 


Let  r—  OP  be  the  radius  vector  drawn  from  an  arbitrary  fixed 
origin,  or  pole,  O  to  a  given  curve ;  on   OP  lay  off  a  length 


=  r'  =  K2/r,    where    tc   is    a  constant;    then   P1    is    said    to 
describe  the  inverse  of  the  given  curve. 

The  theory  of  inversors  is  based  on  the  following  geometrical 
proposition:  if  three  lines  CA=a,  CA'=a,  CO  =  b  (Fig.  65) 
turn  about  C  so  that  O,  A,  A' 
are  always  in  line,  the  product 
OA  •  OA'  remains  constant,  viz. 
OA  •  OA '  =  b2  —  a2.  For  if  the 
circle  of  radius  a  described  about 
C  intersect  the  line  OC  in  B  and 
B1  we  have  OA-OA*  =  OB-OB' 

This  proposition  shows  that  in  the  anti-parallelogram  1234 

(Fig.  66),  with  the  vertex  4  fixed,  the  line  joining  the  vertices  4 

and  2  intersects  the  circle  described  about  3  with  radius  3  2  in  a 

I  point  2r  such  that  2  and  2'  describe  inverse  curves  with  respect 

to  4  as  pole.     For  we  have  4  2^-4  2  =  4  32— 2  32=&2—a2. 


Fig.  66. 


Moreover,  any  parallel  to  42  will  intersect  the  links  41,  43, 
2  i  in  points  O,  A,  A'  dividing  the  three  lines  in  the  same  ratio; 
hence  if  O  be  fixed,  A  and  A'  will  describe  inverse  curves  for 
O  as  pole.  This  is  the  principle  on  which  Hart's  inversor  is 
based. 


148 


KINEMATICS. 


[269. 


269.  Peaucellier's  cell  is  another  inversor  (Fig.  67).  It  con- 
sists of  the  linked  rhombus  A  B  A'  B!  whose  side  we  denote  by 
a,  and  the  two  equal  links  OB,  OB'  of  length  b.  If  O  be  fixed, 
A  and  A'  evidently  describe  inverse  curves  for  O  as  pole. 


Fig.  67. 

The  practical  application  of  inversors  is  based  on  the  property 
that  they  enable  us  to  transform  circular  motion  into  rectilinear 
motion  (see  Art.  271). 

The  inverse  of  a  circle  r=2c  cos#  passing  through  the  pole 
is  a  straight  line  ;  for  we  have  for  the  radius  vector  /  of  the 
inverse  curve  r1  =  K2/r=  K?/2  c  cos  6  ;  hence  /cos  0  =  K2/2  c  which 
is  the  equation  of  a  straight  line  at  right  angles  to  the  polar 
axis,  at  the  distance  /c2/2  c  from  the  pole. 

If  therefore  the  point  A  of  an  inversor  be  made  to  describe 
an  arc  of  a  circle  passing  through  O,  the  point  A'  will  describe 
a  segment  of  a  straight  line.  The  vertex  A  (Fig.  67)  can  beji 
compelled  to  describe  a  circle  by  inserting  the  additional  link; 
O'A  turning  about  the  fixed  point  O'.  If  O'  be  selected  so  as 
to  make  O'O  =  OfA,  say  =  £,  the  circle  described  by  A  will  pass 
through  O\  and  the  motion  of  A'  will  be  confined  to  the 
straight  line  A'D  perpendicular  to  OOf,  at  the  distanc^ 
OD  =  (t>2-a2)/2cfrom  O. 

The  linkage  has  thus  become  a  linkwork,  OO'  being  the  fixed 
link. 


2;i.]  PLANE   MOTION. 

270.  To  determine  the  linear  velocity  v  of  A'  along  DA' 
when  the  angular  velocity  co  of  the  link  OB  is  given,  we  notice 
that  the  instantaneous  centre  C  of  the  link  BA'  lies  at  the 
intersection  of  OB  with  the  line  drawn  through  A'  parallel  to 
OO'.  Let  a)r  be  the  angular  velocity  of  BA'  about  C.  Then 
v  =  a)'  -  CA'  ;  also  since  the  point  B  describes  a  circle  about  O, 
a)l>=a)'  •  CB  \  hence 

CA'  , 


If  BA'   intersect    OO'  in  E,  we  have  from  similar  triangles 
CA':CB=OE\OB;  hence 

v=w  OE. 

The  variable  length  OE  depends  on  the  angles  EOB  —  Q  and 
=  (f>twhich.  are  connected  by  the  relation  (Art.  269) 


The  figure  gives  OE=&cos  6  +  b  sin  6  cot  $  ;  hence,  finally, 
v=wb  sin  #(cot  #  +  cot  </>). 

271.  In  the  steam  engine  and  other  machines  mechanisms 
are  required  for  transforming  the  alternating  rectilinear  motion 
of  the  piston  into  the  reciprocating  circular  motion  of  a  crank, 
eccentric,  or  beam  ;  a  mechanism  of  this  kind  is  called,  rather 
inappropriately;  a  parallel  motion.  The  problem  of  effecting  this 
transformation  has  been  solved  in  various  ways.  Peaucellier's 
inversor  (1864)  was  the  first  accttrate  solution.  Generally,  an 
approximate  solution  is  sufficient  for  practical  purposes.  The 
most  common  of  such  approximations  is  Watt's  parallel  motion. 
This  mechanism  is  a  combination  of  a  linked  parallelogram 
with  a  four  bar  linkwork  with  crossed  links. 


150 


KINEMATICS. 


[271. 


To  fix  the  ideas,  let  4  I  (Fig.  68)  be  the  horizontal  middle 
position  of  the  beam  of  a  beam  engine;  4  is  fixed  and  i  describes 
an  arc  of  a  circle  of  radius  4  I  =#.  We  might  place  a  counter- 
beam  3  2  of  equal  length  turning  about  the  fixed  end  3  so  as  to 
be  in  its  middle  position  parallel  to  4  I  and  so  as  to  make  the 
connecting  link  I  2  nearly  vertical.  The  middle  point  of  i  2 
would  then  describe  a  looped  curve  whose  central  portion  does 


Fig.  68. 


not  differ  very  much  from  a  straight  line;  connecting  this 
middle  point  with  the  piston  rod,  the  problem  would  be  solved. 
But  the  introduction  of  the  large  counter-beam  3  2  in  the 
position  indicated  above  would  be  very  inconvenient.  To  reduce 
the  size  of  the  mechanism  the  counter-beam  3  2  is  placed  nearer 
to  41,  into  the  position  3^2',  and  the  parallelogram  1567  is 
introduced,  the  piston  rod  being  attached  at  7.  Owing  to  the 
property  of  the  linked  parallelogram  (Art.  266),  the  point  7  has 
a  motion  similar  to  that  of  the  point  of  intersection  of  4  7  with 
5  6 ;  it  describes  therefore  approximately  a  straight  line.  The 


273-J  PLANE   MOTION.  !5! 

point  of  intersection  of  4  7  with  5  6  can  be  used  to  connect  with 
the  pump  rods  of  the  engine. 

7.    ACCELERATIONS    IN    THE    RIGID    BODY. 

272.  To  find  the  accelerations  of   the   various   points  of   a 
rigid   body   we   must    compare   the   velocities  of   these  points 
•during  two  consecutive  elements  of   time ;  the  change  of  the 
velocity  divided  by  dt  gives  the  acceleration. 

In  the  case  of  translation  (Art.  250)  the  accelerations  of  all 
points  of  the  body  are  evidently  equal  so  that  the  acceleration 
•of  any  point  may  be  called  the  acceleration  of  the  body. 

273.  In  the  case  of  rotation  about  a  fixed  axis  /,  any  point  P 
of  the  body  at  the  distance  r  from  the  axis  describes  during  the 
element  of  time  dt  a  space  element  ds  —  rdQ  =  wrdt  proportional 
to  this  distance  r,  where  a)  =  d9/dt  is  the  angular  velocity  of  the 
body  about  the  axis  /.     The  linear  velocity  of  P  is  v  =  cor.     The 
:space  element  ds'  described  during  the  next  element  of  time  is 
an  infinitesimal  arc  of  the  same  circle  of  radius  r,  i.e. 

ds'  =  rdO'  =  r(w  +  dw)  dt. 

Drawing  from  any  point  O  (Fig.  69)  the  vectors  OV=ds/dt, 
>OV  =  ds'/dt,  and  resolving  the  ele- 
mentary acceleration  VV[  parallel  to 
the  tangent  and  normal  of  the  path    V 


into  TV  =  dv  =  rda>  and  VT=vdO  = 

ratdO  —  ra)2dt,  we  find  the  tangential 

and  normal  components  of  the  accel-  Fig<  69 

oration  of  P   by  dividing  these  ele- 

ments by  dt.      Hence  denoting  the  angular  acceleration  d<o/dt 

by  a,  we  have 

(0 


The  total  acceleration  of  P, 

(2) 


!^2  KINEMATICS.  [274. 

is  therefore  proportional  to  the  distance  r  of  this  point  from 
the  axis,  so  that  the  accelerations  of  all  points  can  be  found 
as  soon  as  that  of  any  one  point  is  known. 

274.  We  proceed  next  to  the  investigation  of  the  accelera- 
tions of  the  various  points  of  a  rigid  body  having  plane  motion. 
The  motion  is  determined  by  that  of  a  plane  section  of  the  body 
parallel  to  the  plane  of  motion,  and  this  consists  in  the  rolling 
of  the  body  centrode  over  the  space  centrode  (Art.  22). 

During  any  element  of  time  dt,  every  point  P  of  the  plane 
section  rotates  with  angular  velocity  co  about  the  instantaneous 
centre  of  rotation  C  which  is  the  point  of  contact  of  the  two 
centrodes.  During  the  next  element  of  time  dt,  the  angular 
velocity  is  co  +  dco,  and  the  centre  of  rotation  has  changed  to  the 
infinitely  near  point  C^  on  the  space  centrode,  which  has  now 
become  the  point  of  contact  of  the  two  centrodes.  The  accel- 
eration of  a  point  P  at  the  distance  r  from  C  evidently  depends 
on  this  distance  r,  the  angular  velocity  co,  the  angular  accelera- 
tion cx.=da)/dt,  and  the  element  CCl  =  dcr  of  the  space  centrode. 
This  element  divided  by  dt  may  be  regarded  as  a  velocity, 
u  =  d(T/dt,  viz.  the  velocity  with  which  the  instantaneous  centre 
changes  its  position.  We  may  call  it  the  velocity  of  rolling  of 
the  body  centrode.  The  change  in  the  state  of  motion  during 
two  consecutive  elements  of  time  depends  on  a  and  u. 

275.  The  relation  of  the  velocity  of  rolling  u  to  the  angular 
velocity  co  depends  on  the  relative  curvature  of  the  centrodes. 
c,c'. 

To  fix  the  ideas  imagine  these  curves  to  lie  on  the  same  side 
of  their  common  tangent ;  let  da,  da!  be  their  angles  of  contin- 
gence,  and  let  p,  p'  be  their  radii  of  curvature  (Fig.  70). 

The  rotation  about  C  brings  the  second  element  of  c'  to  co- 
incidence with  the  second  element  of  c.  The  angle  dO  of  this 
rotation  is  therefore  equal  to  the  difference  of  the  angles  of 
contingence  of  the  two  curves,  i.e. 


2750 


PLANE   MOTION. 


153 


This  angle  is  therefore  called  the  angle  of  relative  contingence; 
the  quotient  dO/da-  =  (da'  —  da)/ da,  where  da=CCv  is  called 
the  relative  curvature,  and  the  reciprocal  value,  da/d6,  is  the 
radius  of  relative  curvature. 

Nowo>  =  d$/^,  u  =  da/dt;  hence 

a)  _dd  __da'  —  da 
u      da          da 

or  as  da  I  da  =  l/p,  da1  /da=  i/pf, 

co      dO       I        I 


u      da- 


(3) 


i.e.  the  ratio  of  the  angular  velocity  to  the  velocity  of  rolling  is 
equal  to  the  relative  curvature  of  the  centrodes. 


Fig.  70. 

When  p<p',  that  is  when  da>da',  the  relative  curvature  is 
negative.  When  the  centrodes  lie  on  opposite  sides  of  the 
common  tangent  we  should  find  in  absolute  value  dd  =  da' +  da. 
But  taking  into  account  the  sense  of  the  angles  da,  da'  we  still 
have  d0  =  da' —  da.  The  formula  (3)  holds,  therefore,  generally 
if  the  radius  of  curvature  p  of  c  be  taken  as  positive  or  nega- 
tive according  as  it  lies  on  the  same  side  of  the  common 
tangent  with  the  radius  of  curvature  pf  of  c',  or  on  the  opposite 
side. 


I54  KINEMATICS.  [276. 

276.  To  determine  the  components  of  the  acceleration  of  any 
point  P  of  the  body,  it  will  be  convenient  to  imagine  the  angular 
velocities  represented  by  their  rotors :  the  velocity  <a  about  C  by 
a  line  of  length  o>  erected  at  C  at  right  angles  to  the  plane  of 
motion,  on  that  side  of  this  plane  from  which  the  rotation 
appears  counter-clockwise;  similarly  the  angular  velocity  w  +  da 
by  a  parallel  line  of  length  o>  +  da  erected  at  Cv 

The  rotor  a)  +  dco  through  6\  can  be  replaced  by  a  parallel 
rotor  of  the  same  magnitude  and  sense  through  C,  in  combination 
with  a  rotor-couple  whose  moment  is  (co  +  da))-  CCl  =  codo-  (see 
Arts.  255,  256).  This  couple  being  equivalent  to  a  vector  wda- 
at  right  angles  to  the  plane  of  the  couple  produces  an  infinitesi- 
mal velocity  of  translation. 

Thus  the  body,  during  the  first  element  of  time  dt,  rotates 
about  the  axis  through  C  with  angular  velocity  o> ;  and  during 
the  second  element  of  time  dt,  it  can  be  regarded  as  having  the 
angular  velocity  co  +  dco  about  the  same  axis,  and  at  the  same 
time  a  velocity  of  translation  codo-  at  right  angles  to  the  tangent 
at  C.  The  change  in  the  state  of  motion  consists,  therefore,  in 
the  angular  acceleration  d(o/dt=a  and  in  the  linear  acceleration 
wdo-/dt=wu,  the  former  being  due  to  the  change  in  the  magni- 
tude of  the  acceleration,  the  latter  to  the  change  in  the  position 
of  the  axis  of  rotation. 

While  the  acceleration  of  translation  cou  is  the  same  for  all 
points  of  the  figure,  the  angular  acceleration  a  produces  in 

every  point  P  (Fig.  71)  a  linear 
acceleration  proportional  to  its  dis- 
tance r=CP  from  the  centre  C, 
just  as  in  the  case  of  rotation 
about  a  fixed  axis  (Art.  273). 
Resolving  this  acceleration  into 
ci  c  its  tangential  and  normal  compo- 

nents we  have  for  the  acceleration 

of  P  the  following  three  components  :  ar  at  right  angles  to  CPt 
coV  along  PC,  and  atu  at  right  angles  to  CC^. 


277-]  PLANE   MOTION.  !55 

277.  Another  important  method  for  finding  the  components 
of  the  acceleration  of  any  point  P  of  the  body  consists  in 
resolving  (according  to  Art.  254)  the  rotor  w  +  dw  through  £\ 
into  two  parallel  rotors,  &>  through  C,  and  du>  through  a  point  H 
(Fig.  72)  on  the  tangent  CCl  whose  distance  CH=h  from  C  is 
given  by  the  relations 

CC1==C1H=    CH 
dw         a)        &)  +  da) 

Putting  again  CC^dcr,  da-/dt=u,  du>/dt=cx,)  we  find  for  the 
distance  CH=h : 

k  =  ™.  (4) 

a 

The  body  can  therefore  be  regarded  as  having,  during  the 
second  element  of  time  dt,  the  same  angular  velocity  w  about 
the  same  axis  throu-gh  C  as  during  the  first  element  of  time, 
but  in  addition  an  angular  velocity  dw  about  a  parallel  axis 
through  H.  As  the  magnitude  of  the  angular  velocity  about 
C  does  not  change,  the  rotation  about  C  produces  at  any  point 
P  (Fig.  72)  only  a  normal  acceleration  o>V  towards  C,  but  no 


H  h  Ci     C 

Fig.  72. 

tangential  acceleration.  The  infinitesimal  angular  velocity  day 
.about  H,  on  the  other  hand,  produces  only  a  tangential  acceler- 
ation ar1,  perpendicular  to  HP=r1. 

The  acceleration  of  any  point  P  can  therefore  be  resolved 
into  two  components,  one  o>V  directed  towards  the  centre  of 
rotation  C  and  proportional  to  the  distance  r  from  this  centre, 


I56  KINEMATICS.  [278. 

the  other  arr  perpendicular  and  proportional  to  the  distance  rr 

of  P  from  a  point  H  on  the  tangent  at  C,  such  that  CH=  wu/a. 

The  point  H  may  be  called  the  centre  of  angular  acceleration. 

278.  The  resolution  of  the  acceleration  given  in  the  last  article 
enables  us  to  show  the  existence,  at  any  time  t,  of  a  point 
having  at  this  instant  no  acceleration.  This  point  is  called  the 
instantaneous  centre  of  acceleration;  we  shall  denote  it  by  the 
letter  7,  and  its  distances  from  C  and  //",  respectively,  by  r0 
and  rQf. 

For  a  point  of  acceleration  zero  the  components  arf  and  co2r 
must  be  equal  and  opposite.  Now  it  is  evident  that  these 


Fig.  73. 

components  fall  into  the  same  straight  line  only  for  points 
whose  radii  vectores  r,  r'  are  at  right  angles.  The  centre  / 
must  therefore  lie  on  the  circle  described  over  CH  as  diameter 
(Fig.  73).  In  addition  to  this  the  radii  vectores  of  /  must  fulfil 

the  condition 

a>\  =  arQ'.  (5) 

The  locus  of  all  points  for  which  at  any  instant  the  ratio  r/r1  is 
constant  and  equal  to  a/co?  is  a  circle  whose  centre  lies  on  CH 
and  whose  intersections  A,  A'  with  CH  divide  this  distance 
internally  and  externally  in  the  ratio  «/o>2. 

The  two  circles  intersect  in  two  points ;  but  only  for  one  of 


279-]  PLANE   MOTION. 


157 


these  have  the  components  c*2r  and  ar*  opposite  sense.  There 
exists  therefore  only  one  centre  of  acceleration  /,  and  its  radii 
vectores  satisfy  the  conditions 


279.  The  appropriateness  of  the  name  centre  of  acceleration 
for  the  point  /  appears  in  particular  when  the  acceleration  of 
any  point  P  is  referred  to  this  point  7.  For  it  can  be  shown 
that,  if/  be  the  distance  of  P  from  /,  the  acceleration  of  P  can 
be  resolved  into  two  components,  one  o>2p  along  PI,  the  other  ap 
at  right  angles  to  IP  (Fig.  74),  similarly  as  in  the  case  of  rota- 
tion about  a  fixed  axis  (see  Art.  273). 


74. 


To  prove  this  we  resolve  the  component  eoV  of  the  acceler- 
ation of  P  along  PI  and  parallel  to  IC\  it  appears  from  the 
figure  that  these  components  are  aPp  and  &>V0.  The  other 
component  arf  of  the  acceleration  of  P  is  due  to  the  infini- 
tesimal.angular  velocity  dw  about  H.  Replacing  this  dw  about 
H  by  an  equal  angular  velocity  du>  about  /  in  combination  with 
the  infinitesimal  velocity  of  translation  r0Wo>  at  right  angles  to 


158 


KINEMATICS. 


[280. 


HI,  we  obtain,  in  the  place  of  ar',  the  components  ap  at  right 
angles  to  IP  and  ar0'  perpendicular  to  rQf. 

As  of  the  four  components  o>2/,  ap,  o)V0,  arQ'  the  last  two  are, 
by  (6),  equal  and  opposite,  it  follows  that  the  acceleration  of 
P  has  only  the  two  components,  aPp  along  PI,  and  ap  perpen- 
dicular to  IP. 

280.  The    total    acceleration   of    any   point  P   is  therefore 
proportional  to  the  distance  /  of  this  point  from  the  centre  of 
acceleration  /,  viz. 

j=p  V<*2  +  &>4;  (7) 

and  the  angle  -ty  it  makes  with  this  distance  IP,  being  given  by 
the  relation 

tan^  =  -^2,  (8) 

is  the  same  for  all  points.     By  (5),  this  angle  i|r  is  equal  to  the 
angle  CHI. 

All  points  oft^a  circle  described  about  /  as  centre  have 
accelerations  of  equal  magnitude  but  of  different  directions. 
All  points  on  a  straight  line  drawn  through  7  have  accelerations 
that  are  parallel  but  differ  in  magnitude. 

281.  Returning   to   the   resolution  of   the   acceleration   into 

three  components  coV,  ar,  wu, 
as  given  in  Art.  276,  let  us  take 
the  common  tangent  of  the 
centrodes  as  axis  of  x,  their 
normal  as  axis  of  y  (Fig.  75), 
and  let  x,  y  be  the  co-ordinates 
of  any  point  P  whose  distance 
from  C  is  CP  —  r. 

As  the  direction  cosines  of 

Fig.  75.  wV,    ar,    <*u   are   respectively 

-x/r,  —y/r\   —y/r,x/r;  o,  I, 

we  have  for  the  components  of  the  acceleration  /  parallel  to 

the  axes : 


284-]  PLANE   MOTION. 


— 


Jy  =  —  ay  +  ax+  cou. 

The   co-ordinates  x^  yQ  of   the  centre  of  acceleration  /  must 
fulfil  the  conditions 


o,  (10) 

whence  *»  =  /^'  Jo=~/^'  (ll>1 

The  equations  (10)  evidently  represent  the  two  lines  CI  and  HI. 

282.  Let  %=x-xQt  77=7-^/0  be  the  co-ordinates  of  P  with 
respect  to  parallel  axes  through  /;  then,  combining  (10)  and 
(9),  we  find 


These  expressions  show  that  the  total  acceleration/  of  P  is 


since  Vf2+7?2=/=//>,  as  in  Art.  280. 

283.   The  tangential  and  normal  components  of  the  accelera- 
tion j  are  readily  obtained  from  Fig.  74,  as  follows  : 


.?,  A-«v— *  ds> 


The  loci  of  the  points  having  only  normal,  and  only  tangential, 
acceleration  at  any  moment  are  therefore  the  circles  : 


uy  =  Q.  (14) 

284.   Exercises. 

(i)  A  wheel  of  radius  a  rolls  on  a  straight  track.  Find  the  centre 
of  angular  acceleration  H,  (a)  when  the  velocity  v  with  which  the  axis 
of  the  wheel  moves  along  the  track  is  constant  ;  (&)  when  v  is  uniformly 
accelerated  as  when  the  wheel  rolls  down  an  inclined  plane  ;  (c)  when  v 
is  uniformly  retarded,  as  in  rolling  up  an  inclined  plane. 


KINEMATICS.  [284. 

(2)  Show  that   <au  is   the  total   acceleration  of  the   instantaneous 
centre  C. 

(3)  Show  that  the  points  of  the  semi-circle  described  over  CH  as 
diameter  and  containing  /  have  no  tangential  acceleration,  and  that  for 
points  without  the  circle  about  CH  the   velocity   is   increasing  while 
for  points  within  it  is  decreasing. 

(4)  Find  the  locus  of  the  points  of  equal  tangential  acceleration. 

(5)  Show  that  the  locus  of  the  points  having  no  normal  acceleration 
at  a  given  instant  is  a  circle  touching  the  common   tangent   of    the 
centrodes  at  C  and  passing  through  /.     This  circle  is  called  the   circle 
of  inflexions ;  give  the  reason  for  this  name. 

(6)  Find  the  locus  of  the  points  having  equal  normal  acceleration. 

(7)  Show  that  the  diameter  of  the  circle  of  inflexions  is  equal  to  the 
radius  of  relative  curvature  of  the  centrodes. 

(8)  Determine  the  locus  of  the  points  whose  acceleration  at  any 
instant  is  parallel  (a)  to  the  common  normal,  (b)  to  the  common  tangent 
of  the  centrodes. 


286.]  MOTION    IN    THREE   DIMENSIONS.  !$! 

IV.     Solid  Kinematics. 

I.       MOTION    OF    A    POINT    IN    A    TWISTED    CURVE. 

285.  We  have  so  far  considered  only  those  cases  of  motion 
where  the  path  of  the  point  is  a  plane  curve.  In  the  most  gen- 
eral case  when  the  path  is  a  so-called  twisted  or  tortuous  curve 
we  may  refer  it  to  three  rectangular  axes  and  resolve  the  veloc- 
ity v  as  well  as  the  acceleration  j  each  into  three  rectangular 
components  parallel  to  these  axes  : 


dx  •       •  dvr 

-,  j,=Jcoa^-  =  —, 


.-§, 


dz  .        .  dvf      d^z 

-,  ,;=/«»„  =  -•=—, 


i) = V  ^ 2  -h 


286.  As  polar  co-ordinates  of  the  point  P  we  take  the  radius 
vector  OP  —  r,  the  colatitude  xOP  =  0,  and  the  longitude 

Q  =  $,  Q  being  the  projection  of  P  on  the  plane  yOz 
(Fig.  76). 

The  velocity  z;  can  be  resolved  into  three  rectangular  compo- 
nents :  vr  along  r,  v9  at  right  angles  to  r  in  the  plane  x  OP  of 
the  angle  0,  and  v^  at  right  angles  to  this  plane.  To  find  their 
values  we  take  the  element  PP'  =  ds  of  the  curve  described  by 
the  point  P  as  diagonal  of  an  infinitesimal  parallelepiped  having 
its  edges  in  those  three  rectangular  directions.  The  three 

PART   I — II 


1 62 


KINEMATICS. 


[287. 


edges  concurring  in  P  are  evidently  dr,  rd6,  rsin0d<f);  hence 
the  components  of  the  velocity  are 


dr 
'  dt' 


dt 


X 


Fig.   76. 

287.  The  components  of  the  acceleration  /  in  polar  co-ordi- 
nates are  readily  obtained  by  considering  that  the  accelerations 
of  the  point  P  in  the  direction  at  right  angles  to  Ox  in  the 
plane  xOP  and  in  the  direction  at  right  angles  to  this  plane 
are  the  same  as  the  accelerations  of  the  point  Q  (Fig.  76);  they 
are  therefore,  by  Art.  161,  (6),  since  RP=OQ  = 


in  the  direction  RP,  and 
i 


r  sin  6  dt 


dt 


at  right  angles  to  the  plane  of  the  angle  0.  The  component  of 
j  parallel  to  the  axis  Oz  is,  of  course,  d\r  cos  0)/dt2,  Resolv- 
ing these  three  components  parallel  to  the  three  rectangular 
directions  along  r,  at  right  angles  to  r  in  the  plane  xOP,  and 


289.]  MOTION    IN    THREE   DIMENSIONS.  163 

at  right   angles  to  this   plane,   and   collecting   the  terms,  we 
obtain  : 


.    fidr  d$> 
dt*   '  dt  dt1  '"dt   dt 

288.  It  is  to  be  noticed  that  the  resolution  of  the  accelera- 
tion/into  a  tangential  component/  and  a  normal  component/,, 

•  _^       • _  ^ 
dt'  p' 

given  in  Art..  159,  holds  for  twisted  curves  as  well  as  for  plane 
curves,  provided  the  normal  be  understood  to  mean  the  prin- 
cipal normal  of  the  curve,  and  p  the  radius  of  absolute  curvature 
at  P.  For  it  follows  from  the  definition  given  in  Art.  155  that 
the  acceleration  lies  in  the  plane  of  the  tangent  and  principal 
normal  at  P,  so  that  the  component  along  the  binormal  is  zero. 

289.  This  can  also  be  seen  from  the  expressions  for  the  com* 
ponents  of/  in  Cartesian  co-ordinates,  jx  =  d*x/dt*,  jy  =  d*y/dt*, 

jg  =d*z/dt*.     For  since  ^=^  £^  etc.,  we  have 

dt     ds  dt 

dx  .  fds\*  d*x 


= 
Js      dt*      dt*  ds      \dt 


=        ^  , 

Jy     dt*     dt*  ds     \dt    ds*' 

.  =  d*z^d*s<te     (ds\*d*z 
Jz     dt*     dt*  ds      \dt)ds* 

Now,  dx/ds,  dy/ds^   dz/ds  are   the   direction   cosines   of  the 
tangent  of  the  curve,  while  pd*x/ds*y  pd*y/ds*,  pd*z/ds*  are.  the: 


KINEMATICS.  [290. 

direction  cosines  of  the  principal  normal.     The  formulae  show 
therefore  that  the  acceleration  j  consists  of  two  components, 

^ !=^  along  the  tangent,  and  L/!^Y«.!?  along  the  normal. 
dt*     dt  p\dt J       p 


2.     VELOCITIES    IN    THE    RIGID    BODY. 

290.  When  the  motion  of  a  rigid  body  is  a  translation,   all 
points  of  the  body  have  at  any  instant  equal  and  parallel  veloc- 
ities (Art.  250).     The  velocity  v  =  ds/dt  of  any  one  point   can 
therefore  be  called  the  velocity  of  the  body.     The  body  can  be 
subjected  at  a  given  instant  to  several  velocities  of  translation, 
and  the  resultant  velocity  is  found  by  the  geometrical  addition 
of  the  vectors  representing  the  component  velocities. 

291.  When  a  rigid  body  rotates  at  the  time  t  about  an  instan- 
taneous axis    /,   all  its   points   (excepting   those   on   the   axis) 
describe   infinitesimal   arcs    of   circles   of    angle    dO,    and    the 
angular  velocity  a>  =  dd/dt  of   any  point  of  the  body  may  be 
called  the  angular  velocity  of  the  body.     This  angular  velocity 
can  be  represented  geometrically  by  its  rotor  o>  laid  off  on  the 
axis  /(Arts.  68,  69,  252). 

As  this  rotor  is  proportional  to  the  infinitesimal  angle  o 
rotation  dd,  the  propositions  proved  in  Arts.  62,  66,  67,  68,  fo 
the  composition  and  resolution  of  infinitesimal  rotations  can  b 
applied  directly  to  angular  velocities.  The  propositions  refer 
ring  to  parallel  axes  have  been  discussed  in  Arts.  254-257. 

292.  If  in  Art.  62  we  divide  equation  (i')  by  dt2  and  divide 
the  denominators  of  equation  (2')  by  dt,  we  obtain 

ft,2  =  0,12  +  ft,22  +  2ft)1ft)2  cos (/^  (i 

sin  (/!/)  _  sin  (//2)  _  sin  (//2)  , 

ft)  2  ft)}  ft> 

The  meaning  of  these  equations  can  be  stated  as  follows.     Le 
a   rigid    body  be    subjected    simultaneously    to    two    angula 


293-]  THE   RIGID    BODY.  ^5 

velocities  about  intersecting  axes,  ^  about  /x  and  <o2  about  /2. 
Represent  these  angular  velocities  by  their  rotors  cov  &>2  laid 
off  on  the  axes  /p  /2  from  their  point  of  intersection  O  and 
construct  their  geometric  sum  w  ;  that  is,  form  the  diagonal 
of  the  parallelogram  whose  adjacent  sides  are  coly  co2.  Then  co 
is  the  rotor  of  the  resulting  angular  velocity. 

This  proposition  is  known  as  the  parallelogram  of  angular 
velocities. 

It  follows  that  the  resultant  of  any  number  of  simultaneous 
angular  velocities  whose  axes  all  intersect  in  the  same  point  is 
a  single  angular  velocity  whose  rotor  is  found  by  geometrically 
adding  the  rotors  of  the  components. 

293.  Conversely,  an  angular  velocity  co  about  an  axis  /  can 
always  be  replaced,  in  an  infinite  number  of  ways,  by  two  (or 
more)  angular  velocities  whose  geometric  sum  is  o>,  about  two 
(or  more)  axes  passing  through  any  point  O  of  /  and  lying  in 
the  same  plane  with  /. 

Thus,  for  instance,  the  angular  velocity  co  about  the  instan- 
taneous axis  /  can  be  resolved  into  three  components  cox,  coy,  wz 
about  three  rectangular  axes  Oxy  Oy,  Oz  passing  through  any 
point  O  of  /,  and  we  have 

a>2  =  <»z2  +  aVJ  +  aVJ.  (3) 

The  linear  velocity  v  of  any  point  P  of  a  body  rotating  with 
angular  velocity  w  about  the  axis  /  can  be  expressed  by  means 
of  the  components  &>x,  coy,  wz  of  co  and  the  co-ordinates  xt  y,  z  of 
the  point  P.  The  component  cox  produces  at  P  a  velocity  whose 
components  along  the  axes  Ox,  Oy,  Oz  are  o,  —  co^,  wxy\  simi- 
larly, o)y  gives  the  components  cOyZ,  o,  —  coyx;  and  coz  gives  —  wzyt 
a>2x,  o.  Hence,  combining  the  terms  that  lie  along  the  same 
axis,  the  components  of  the  velocity  v  of  the  point  P  are 


(4) 


KINEMATICS.  [294. 

294.  If  a  rigid  body  be  subjected  at  the  time  t  to  two  simul- 
taneous angular  velocities  cov  &>2  about  skew  (or  crossing,  i.e. 
not  intersecting  and  not  parallel)  axes  /x,  /2,  or  if  it  be  subjected 
to  an  angular  velocity  co  about  an  axis  /  and  a   simultaneous 
linear   velocity  v   not    perpendicular   to  /,  its  state  of   motion 
during  the  time  dt  cannot  be  expressed  by  a  single  angular  or 
linear  velocity. 

The  body  can  be  said  to  have  in  either  case  a  twist-,  or  screw- 
velocity,  i.e.  an  angular  velocity  co  about  an  axis  /  combined  with 
a  linear  velocity  z/0  parallel  to  this  axis. 

To  prove  this  in  the  latter  of  the  two  cases  it  is  only  necessary 
to  resolve  v  into  a  component  z/0  parallel  to  /  and  a  component 
v'  perpendicular  to  /.  The  latter,  being  equivalent  to  a  rotor 
couple  (&>,  —  &>)  of  moment  v'  =pco  (see  Art.  256),  combines  with 
the  given  angular  velocity  co  about  /  into  an  angular  velocity  co 
about  a  parallel  axis  /'  at  the  distance  p  =  v' /a>  from  /.  The 
combination  of  the  angular  velocity  co  about  /  with  the  simul- 
taneous oblique  linear  velocity  v  is  therefore  equivalent  to  the 
angular  velocity  co  about  V  with  the  simultaneous  linear  velocity 
VQ  parallel  to  /'. 

295.  When   the   rigid  body   has   two   simultaneous   angular 
velocities  a>lt  co2  about  skew  axes  llt  /2,  the  reduction  is  best  made 
by  replacing  co2  about  /2  by  an  equal  angular  velocity  o)2  about  a 
parallel   axis    V   intersecting  tlt  in  combination  with  a  linear 
velocity  v=pco<1  perpendicular  to  the  plane  of  /2  and  /'  (Art.  257). 
The  angular  velocities  co^  about  /:  and  o>2  about  /'  combine  (by 
Art.  292)  into  a  singular  angular  velocity  whose  rotor  is  the 
geometric  sum  of  eoj  and  o>2.     The  case  is  therefore  reduced  to 
the  preceding  one. 

296.  It  follows  from  the  preceding  articles  that  any  number 
of  simultaneous  linear   and  angular  velocities    can    always   be 
combined  into  a  single  twist-velocity  about  the  central  axis. 


298.]  THE    RIGID    BODY. 


3.       ACCELERATIONS    IN    THE    RIGID    BODY. 

297,  The  accelerations  of  the  points  of  a  rigid  body  are 
found  by  comparing  the  velocities  of  these  points  during  two 
successive  elements  of  time. 

If  the  motion  of  the  rigid  body  be  a  pure  translation,  all 
points  of  the  body  describe  equal  and  parallel  curves.  The 
accelerations  of  all  points  being  equal  and  parallel  (Art.  272), 
the  acceleration  j  of  any  one  point  of  the  body  can  be  spoken 
of  as  the  acceleration  of  the  body.  It  can  be  resolved  into  a  tan- 
gential component  jt  along  the  tangent  to  the  path  of  any  point 
and  a  normal  component  /„  along  the  normal  to  the  path,  and  we 
have,  just  as  in  Art.  159, 


(i) 


298.  If  the  motion  of  the  rigid  body  be  a  pure  rotation  about 
the  same  axis  /  for  at  least  two  successive  elements  of  time  dtt 
all  points  describe  arcs  of  circles  whose  centres  lie  on  the  fixed 
axis  /.  As  shown  in  Art.  273,  the  acceleration  j  of  any  point  P 
whose  distance  from  /  is  r  can  be  resolved  into  a  tangential 
component  jt  perpendicular  to  the  plane  (/,  P)  and  a  normal  com- 
ponent /„  at  right  angles  to  the  axis  /;  and  we  have  (Art.  273) 


(2) 


where  «  is   the   angular  velocity  and    a  =  da)/dt  the   angular 
acceleration  of  the  body. 

The  normal  component  /„  being  always  directed  towards  the 
axis  of  rotation  /  is  sometimes  called  the  centripetal  acceleration. 


1 68 


KINEMATICS. 


[299. 


299.  If  the  motion  of  the  rigid  body  consists  in  a  rotation 
about  an  axis  /  during  the  first  element  of  time  and  a  rotation 
about   an    infinitely    near  parallel  axis   /'    during  the    second 
element  of  time,  we  have  the  case  of  plane  motion  of   a  rigid 
body  which  has  been  treated  in  Arts.  274-284. 

It  remains  to  discuss  the  case  of  intersecting  axes,  which  is  of 
fundamental  importance  in  the  kinetics  of  the  rigid  body. 

When  the  axes  about  which  the  body  rotates  in  the  successive 
elements  of  time  intersect  at  a  point  O,  this  point  remains  fixed 
during  the  motion  and  may  be  called  the  centre  of  rotation.  The 
motion  of  a  rigid  body  with  a  fixed  point  may  be  called 
spherical  motion. 

The  accelerations  of  the  points  of  a  body  in  spherical  motion 
can  be  studied  in  a  manner  strictly  analogous  to  that  used  in 
the  case  of  plane  motion  (Arts.  274-284). 

300.  Let  the  body  rotate  during  the  first  element  of  time  dt 
with  angular  velocity  o>  about  an  axis  /,  and  during  the  second 

element  of  time  dt  with  angular  velocity 
ft)  +  ^/&)  about  an  axis  /'  intersecting  / 
in  the  point  O  and  making  with  /  the 
infinitesimal  angle  (/,  l')=dcr.  The 
angular  velocities  can  be  represented 
by  their  rotors,  o>  along  /,  co  +  dco  along 
/'  (Fig.  77). 

The  rotor  w  +  da  along  I'  can  be 
resolved  into  a  rotor  &>  along  /  and  an 
infinitesimal  rotor  d$  along  an  axis  h 
that  passes  through  O  and  lies  in  the 

plane  (/,  /').     The  value  of  d$  and  the  angle  (/,  h)  =  y  are  given 

by  the  relations 

sin  (/,  /')  _  sin  (/',  /z)  _  sin  (/,  K) 
d<f)  a)  a)  +  dot 


Fig.  77. 


(3) 


whence 


sin  (/,  ^)=  sin  7  =  0)—- 


(4) 


302.]  THE   RIGID    BODY. 

Putting  d<rldt=u,  d^/dt—^  we  may,  similarly,  as  in  Art.  274, 
call  u  the  velocity  of  rolling  of  the  cone  of  instantaneous  axes 
and  a  the  angular  acceleration.  With  'these  notations 


301.  The  appropriateness  of  these  names  will  appear  by 
considering  that  the  body  can  now  be  regarded  as  having,  for  two 
successive  elements  of  time,  the  same  angular  velocity  w  about 
the  same  axis  /,  modified  during  the  second  element  of  time  by 
the  additional  infinitesimal  angular  velocity  d<j>  about  the  axis  k, 
which  is  called  the  axis  of  angular  acceleration. 

Thus  the  rotation  about  /  produces  only  centripetal  (and  no 
tangential)  acceleration  which  at  unit  distance  from  /  is  =o>2  and 
is  directed  at  right  angles  to  /  towards  /  (see  Art.  298),  while 
the  rotation  about  h  gives  at  unit  distance  from  h  the  infinitesi- 
mal velocity  d$  at  right  angles  to  the  planes  through  h  and  thus 
produces  the  angular  acceleration  a  =  d$>/dt,  which  may  be 
represented  by  a  vector  a  along  h. 

The  projection  of  d$  on  /  is  evidently  da  (see  Fig.  77),  so 

that 

da)     I  dw  x,x 

cos7=  —=-—-.  (6) 

a(f>     a  dt 

Squaring  and  adding  the  equations  (5)  and  (6),  we  find 


302.  These  results  are  further  illustrated  by  another  resolu- 
tion analogous  to  that  of  Art.  276. 

Imagine  the  body  subjected,  during  the  second  element  of 
time,  to  the  equal  and  opposite  angular  velocities  v  +  dw  and 
—  (<»  +  </&>)  about  /  (Fig.  78);  then  combine  co  +  dco  about  /' 
with  —  (<»  +  dfe>)  about  /  into  the  infinitesimal  angular  velocity 
(o>  -t-ak>)  sin  da  =  coda-  about  an  axis  n  through  O  at  right  angles 


170 


KINEMATICS. 


[302. 


•to  /in  the  plane  (/,  /').    This  is  equivalent  to  resolving  the  rotor 
w  +  do>  along  /'  into  the  rotors  to  +  du>  along  /and  a>d<j  along  ;/. 

The  body  can  now  be  regarded  as  rotating  during  both  ele- 
ments of  time  about  the  axis  /,  viz.  during  the  first  element 
with  angular  velocity  o>,  during  the  second  with  angular  velocity 
w  +  dw,  and  in  addition  to  that  during  the  second  element  about 
the  axis  n  with  the  infinitesimal  angular  velocity  ndcr. 


(n), 


Fig.  78. 

The  rotation  about /(Art.  298)  produces,  for  points  at  unit 
distance  from  /,  a  centripetal  acceleration  o>2  perpendicular  to  /and 
a  tangential  acceleration  dw/dt  which  may  be  represented  by  a 
rotor  dw/dt  along  /.  The  rotation  about  n  gives  to  points  at 
unit  distance  from  n  an  infinitesimal  velocity  wda  at  right  angles 
to  the  planes  through  n  and  thus  produces  an  acceleration 
todvldt^tou  which  may  be  represented  by  a  rotor  along  n.  The 
rotors  dwjdt  along  /  and  mi  along  n  being  at  right  angles  to 
each  other  (see  Fig.  78),  combine  to  form  the  angular  acceler- 
ation 


It  is  apparent  that  the  component  da/di  of  a  has  the  effect 


303.]  THE   RIGID    BODY.  j^j 

of  changing  the  magnitude  of  o>  by  the  amount  da,  without 
affecting  the  direction  of  the  axis,  while  the  effect  of  the  com- 
ponent a)U  is  to  incline  the  axis  /  by  the  angle  dcr. 

303.  To  obtain  analytical  expressions  for  the  components  of 
the  acceleration  of  any  point  P  of  a  rigid  body  in  spherical 
motion,  let  us  take  the  centre  of  rotation  O  as  origin  of  a 
system  of  fixed  rectangular  axes.  Let  x,  y,  z  be  the  co-ordinates 
of  P  \  a,  @,  7  the  direction  cosines  of  the  instantaneous  axis  /; 
and  X,  fji,  v  those  of  the  perpendicular  PQ  =  r  let  fall  from  P  on 
this  axis  /. 

The  total  acceleration  of  P  is  composed  of  the  centripetal 
acceleration  o>V,  which  is  directed  along  PQ,  and  the  component 
arising  from  the  angular  acceleration  a  (Art.  301). 

The  components  of  o>V  along  the  axes  of  x,  y,  z  are  \&>V, 
/xcoV,  vco2r.  Projecting  the  closed  polygon  OQPO  on  each  of  the 
axes,  we  find 


or,  since  OQ  is  the  projection  of  OP  on  /,  i.e.  OQ  = 
\r=  a  (ax 


vr=y  (ax  +  fty  +  yz)  —  z. 

Multiplying  these  equations  by  o>2  and  putting  aco  =  cox,  /3o)  =  coyt 
jo)  =  a)z,  we  find  for  the  components  of  the  centripetal  accelera- 
tion of  the  point  (x,  y,  z)  : 


=  cox  a)x 
G)2/(wx^H-ft>yj+G>^')—  oPy*  (8) 

=  coz  (wxx  -f-  coy  y  +  &>^)  —  aPz. 

The   angular   acceleration  a  =  d$/dt  (Art.    301)  has  for  its 
components  along  the  axes  of  x,  y,  z 

d«>x        _  d(0y        _  dto, 

CC   =  -  -f      **«  —  -  *>      ^z  —  -  • 

*     dt  dt  dt 


KINEMATICS.  [304. 

The  component  ax  produces  an  infinitesimal  angular  velocity 
axdt  about  the  axis  Ox  ;  and  hence  gives  to  P  the  infinitesimal 
velocities  o,  —  axzdt,  axydt  along  the  axes  Ox,  Oy,  Oz  (see  Art. 
293);  similarly,  aydt  produces  the  velocities  ayzdt,  o,  —  ayxdt, 
and  azdt  produces  —  azydt,  azxdt,  o.  Collecting  the  terms  paral- 
lel to  each  axis  and  dividing  by  dt,  we  find  the  components  of 
the  acceleration  of  P  due  to  the  angular  acceleration  a  : 

ayZ-azy,  a,x—a^t  a.£y-ayx.  (9) 

Finally,  combining  the  corresponding  terms  in  (8)  and  (9)  and 
remembering  that  a.x=da>Jdt,  ay=da)y/dt,  az=da)z/dt,  we  find  the 
following  expressions  for  the  components  of  the  total  accelera- 
tion j  of  the  point  P  (x,  y,  z)  : 


> 

at        at 

^*-^s,        (10) 

dcor       dw.. 
—fy  —  -*-x. 
at         at 

304.  The  formulas  (10)  for  the  components  of  the  accelera 
tion  of  any  point  (xy  y,  z)  of  a  body  rotating  about  a  fixed  poin 
O  can  also  be  derived  by  differentiating  the  expressions  (4)  ir 
Art.  293,  which  represent  the  components  of   the  velocity  o 
such  a  point.     It  is  only  necessary,  after  the  differentiation,  tc 
substitute  for  dxjdt,  dy/dt,  dz/dt  their  values  from  (4),  Art 
293,  and  to  remember  that  cc?  =  cox2+a)yz-{-(t)22. 

305.  The  complete  study  of  the  motion  of  a  rigid  body  in  th< 
most  general  case,  in  particular  the  investigation  of  its  accelera 
tions,  is  beyond  the  scope  of  the  present  work. 

In  addition  to  the  works  previously  referred  to,  the  following  work 
on  kinematics  may  here  be  mentioned. 

An  elementary  introduction  to  kinematics,  without  the  use  "of  the  in 
fmitesimal  calculus,  will  be  found  in  J.  G.  MACGREGOR,  An  elementary 


305.]  THE   RIGID    BODY.  I^ 

treatise  on  kinematics  and  dynamics,  London,  Macmillan,  1887.  This 
may  be  supplemented  by  W.  K.  CLIFFORD,  Elements  of  dynamic,  part 
i,  Kinematic,  ib.,  1878.  For  more  advanced  study  see  G.  M.  MINCHIN, 
Uniplanar  kinematics  of  solids  and  fluids,  Oxford,  Clarendon  Press, 
1882  ;  THOMSON  and  TAIT,  Natural  philosophy,  new  edition,  part  i,  ib., 
1879  ;  W.  SCHELL,  Theorie  der  Bewegung  und  der  Kr'dfte,  vol.  i,  1879, 
Leipzig,  Teubner;  J.  SOMOFF,  Theoretische  Mechanik,  ubersetzt  von 
A.  Ziwet,  part  i,  Kinematik,  Leipzig,  Teubner,  1878  ;  E.  BUDDE,  Allge- 
meine  Mechanik  der  Punkte  und  starren  Systeme,  Berlin,  Reimer,  1890  ; 
H.  RESAL,  Traite  de  cinematique  pure,  Paris,  Mallet-Bachelier,  1862; 
E.  BOUR,  Cours  de  mecanique  et  machines,  part  i,  Cinematique,  2d  ed., 
Paris,  Gauthier-Villars,  1887;  E.  COLLIGNON,  Traite  de  mecanique, 
part  i,  Cinematique,  3d  ed.,  Paris,  Hachette,  1885  ;  E.  VILLIE,  Traite 
de  Cinematique,  Paris,  Gauthier-Villars,  1888. 


ANSWERS. 


Page  17. 

(i)    Join  the  point  P  to  the  instantaneous  centre  (7;  the  direction 
'  of  motion  is  perpendicular  to  CP. 

(3)  See  Art.  29. 

(4)  See  Art.  29.     With  O  as  origin  and  a  parallel  to  /as  axis  of  yy 
_the  fixed  centrode  is  (y2—cxy=a2(x2 +y2),  where  a  is  the  radius  of  the 

circle  about  <9,  and  c  the  distance  of  O  from  /. 

(6)  The  fixed  centrode  is  a  circle  passing  through  O',  O"  ;  the  body 
centrode  is  a  circle  of  twice  the  radius.  The  path  of  any  point  in  the 
fixed  plane  is  in  general  a  limacon  of  Pascal. 

(8)  Consider  the  initial  and  final  positions  of  the  point  of  intersec- 
tion of /0  and  /j. 

Page  28. 

(i)    24  miles;  E.  35°  S. 

(3)  V3- 

(4)  10.7  miles;  E.  14!°  S. 

(6)  2  a  cos  (<*/2). 

(7)  (a)  120°;   (£)  i6o°48'.6. 

(10)    Inclination    to    vertical:     (a)    n°.3j    (<£)    2i°.8;     (<r)    45°; 
(d)  67°.4. 

(n)    iojft.j  "=2471°. 

175 


176  ANSWERS. 

(16)    On   PoPlt  P0P2    construct    the    parallelogram    P0P^P2   and 
draw  P{S  parallel  to  QP2,  S  being  the  intersection  with  the  diagonal  P^R. 

(18)    Apply  (16). 

Page  39. 

(i)    For  the  angle  6  of  the  resulting  rotation  we  have  sin  (0/2) 
=  J-V5/2  ;    for  the   position   of  its   axis   /,   sin(4/)  =  2/Vs,   sin(//2) 

-Vtfc 

(3)      22°. 

Page  47. 

(1)  (a)  41  miles  per  hour ;   (c)  19.1,8.2;   (d)  10  h.  19  m. 

(3)  At  2  h.  22  m. ;  203  miles  from  Detroit. 

(4)  (a)  5.9;   (2)  40.6;   (c)  44;   (/)  35.25;   (e)  1093. 

(5)  15- 

(7)  185,000  miles  per  second. 

(8)  (a)  ih.;   (b)  15  m. 

(9)  3of . 

(I0)    37i  miles  per  hour. 

Page  53. 
(0    H- 

(2)  32.186. 

(3)  Nearly  yL  ft.  per  second  per  second. 

(4)  0.0034. 

Page  56. 

(1)  (a)  96.6;   (£)  402.5;   (c)  144.9. 

(2)  0.275. 


(4)    h  =c\  / H -y -f  2  /  4- -  J    ;  an  approximate  value  is 

h  = %- —  •    For  a  direct  numerical  computation,  the  method 

*('+**) 

of  successive  approximations  may  be  used.     Thus,  neglecting  the  time 


ANSWERS. 

/,  required  by  the  sound,  find  the  depth  s  approximately  from  s  = 
with  /  =  4  ;  with  this  value  of  s  find  /2  ;    hence  the  time  of  fall  /b  with 
which  correct  s  ;  etc.     Result  :  s  =  70.4  metres. 

(5)    (a)  4  min.  ;  (b)  1  1/60  ;  (c)  30  miles  per  h.  ;  (d)  after  3  m.  28  s. 

(8)  (a)  40,000  ft.;    (b)  ±715.5  ft.  per  second;   (c)  i  m.  40  s.  ; 
{d)  1600  ft.  per  second;  (e)  i  m.  12.4  s.  and  27.6  s. 

(9)  80  ft.  per  second. 

(10)    (a)  t=h/vQ-}  (b)  h-s  = 


Page  60. 

(1)  (a)   26,000  ft.  per  second;  (b)  34  m.  48  s. 

(2)  It  represents  a  cycloid. 

(4)  v^R/(2gR  —  002).     If  v}  5  2gR,  the  particle  will  not  fall  back. 

(5)  Height  =  >?;     time    of    ascending  =  ^|—f  i  +-j  =  time    of 

o     \  / 

falling  back  =  34  m.  48  s.  ;  hence  whole  time  =  i  h.  9  m.  36  s. 

(6)  7  miles  per  second. 

Page  63. 

(2)  v  =  26,000  ft.  per  second  ;  /=  i  h.  25  m.  4.5  s. 

(3)  2S  =  R(#*  +  e-v*)t  or  s  = 


Page  65. 

(i)    lira  #=£///,  for  lim  /=  oo. 

v*  CQS  Vg^  ^  —  ^  sin 


(4)    Time  of  ascent          T=     I__t3Lirl\  -v0 ; 

V**          X<^ 

height  of  ascent       ^  =  ^7  log  f  i  4-  -  »02  )• 


PART   I — 12 


T^8  ANSWERS. 

(5)  Compare  the  height  of  ascent  in  Ex.  (4)  to  the  distance  fallen 
through  as  obtained  in  (27),  Art.  126.  If  vl  be  the  velocity  with  which 
the  particle  returns  to  the  starting  point,  we  find 


(6)  v  =  v0e-kt, 

(7)  ^  =    (i- 


Page  69. 

(i)    w  =  TT  radians  ;  v  =  18.8  ft.  per  second. 

W  (*)3i;  (J)3»- 

(3)  -0.157- 

(4)  5- 

(5)  (a)  402.1  ;  (b)  25.1  seconds. 

Page  73. 

(1)  r  =  v$t,  6  =  <o/;  hence  r  =  v^B/a,  a  spiral  of  Archimedes. 

(2)  About  the  pole  O  describe  a  circle  of  radius  a  and  find  its 
intersection  Q  with  the  perpendicular  to  the  radius  vector  OP  drawn 
through  O  ;  then  QP  is  the  normal.     Proof  by  Ex.  (  i  )  . 

(3)  For  the   direction  of    v   see   Art.    31,    Ex.    (2).      Resolving 
v  into  v0  parallel  to  the  track  and  vl  along  the  tangent  to  the  wheel, 
it  appears  that  v  bisects  the  angle  between  these  components  ;   hence 
v  =  2  v0  cos  CAP,  where  C  is  the  centre  of  the  wheel,  and  A  its  lowest 
point. 

(5)  For  the  ellipse,  r  +  r*=  const.  ;  hence  —  =  —  —  ,  />.  the  pro- 

at  at 

jections  of  the  velocity  on  the  radii  vectores  are  equal. 

(6)  The  projections  of  the  velocity  on  the  radius  vector  and  on 
the  focal  axis  are  in  the  constant  ratio  e  of  the  focal  radius  vector 
to  the  distance  to  the  directrix.     It  follows  that  the  tangent  intersects 
the  directrix  in  the  same  point  as  does  the  perpendicular  to  the  radius 
vector  through  O. 

(7)  40. 


ANSWERS.  i;9 

(10)    vl=  20  ft.  per  second,  nearly;  angle  =  20^°. 

(  1  1  )    The  relative  velocity  of  P2  with  respect  to  J\  must  always  pass 
through  Plt     The  locus  of  Q  is  a  circle. 

(12)  A  cycloid. 

(13)  About  20". 

Page  79. 

(4)  233!  ;  24!  ;  933^  ft.  per  minute. 

(5)  16.6  knots;  560  ft.  per  minute. 

(6)  55°;  66°;  2|  in. 

(7)  0.174,  0.119,  0-146  of  the  stroke. 

Page  85. 


(2)  By(2),Art.  i59,/n 

(3)  By  Art.  159,  jn  —j  sin  i//  =  v*/p  ;  hence  v2  =/  •  p  sin  if/. 

(4)  Since  j  is  directed  towards  A,  taking  A  as  origin,  we  have 

JQ  =  o,  i.e.  r-—  =  const.  ;  comp.  Art.  135. 
«/ 

(5)  —  =  eo  =  const.,  r  =  const.  ;  hence,  by  (6),  Art.  161,  j=jr 
at 

=  -r<J. 

(6)  /=no2. 

Page  86. 

(i)    (a)  1718  ft.  above  the  point  ;  (£)  after  2  m.  52  s.  ;  (c)  1891  ft.  ; 
(d)  21.2  miles  an  hour. 

(4)    45°- 

(7)  Construct  a  circle  having  the  given  point  as  its  highest  point 
and  touching  (a)  the  straight  line,  (b)  the  circle. 

Page  90. 

(9)    (a)  1  74ft.;   (b)  in  about  8  seconds;   (c)  254  ft.  per  second, 
inclined  at  an  angle  of  about  5°  to  the  vertical. 

(10)    227  ft.  per  second. 


PART  II: 
INTRODUCTION  TO  DYNAMICS;  STATICS 


PREFACE. 


THE  subject  of  statics  is  here  developed  only  so  far  as  is  absolutely 
necessary  in  order  to  lay  the  foundation  on  the  one  hand  for  the  study 
of  elementary  kinetics,  on  the  other  for  applied  mechanics.  From 
the  former  point  of  view  it  was  desirable  to  bring  out  clearly  the  con- 
nection of  the  subject  with  the  general  science  of  mechanics  and  to 
determine  its  place  as  a  subdivision  of  the  larger  science.  The  second 
section  of  Chapter  III  should  be  considered  only  as  preliminary ;  the 
fundamental  laws  of  dynamics  can  of  course  be  fully  understood  only 
by  studying  kinetics.  Prominence  is  given  throughout  to  geometrical 
methods  and  graphical  constructions  because  these  seem  to  conform 
best  to  the  nature  of  the  subject.  The  applications  given  here  and 
there  are  to  be  regarded  merely  as  illustrations  of  the  general  prin- 
ciples. 

The  following  articles  might  be  omitted  at  first  reading :  18,  19,  20, 
34,  43,  44,  48,  52,  113,  117-127,  152-164,  180,  181,  209,  210,  214, 
220-225,  257~2^5- 

ALEXANDER  ZIWET. 

ANN  ARBOR,  MICH., 
October,  1893. 


CONTENTS. 

CHAPTER   III. 
INTRODUCTION  TO  DYNAMICS. 

I.    MASS;   MOMENTS  OF  MASS;  CENTROIDS: 

1 .  Mass  ;  density 1 

2.  Moments  and  centres  of  mass 4 

3.  Examples  of  the  determination  of  centroids .         .         .  u 

II.    MOMENTUM;   FORCE;   ENERGY 34 

CHAPTER   IV. 
STATICS. 

I.    INTRODUCTION 45 

II.    CONCURRENT  FORCES 54 

III.  PARALLEL  FORCES    .        .        . 59 

IV.  THEORY  OF  COUPLES        .        .    |- 76 

V.    PLANE  STATICS  : 

1.  The  conditions  of  equilibrium        .        .         .         .         -83 

2.  Stability.         .         .         .  ,v      .  .-      .         .         .92 

3.  Jointed  frames         .         .    .  ...     ..  •         •      99 

4.  Graphical  methods IO3 

5.  Friction  .         .         . 109 

vii 


Vlll 


CONTENTS. 


VI.    SOLID  STATICS: 

1.  The  conditions  of  equilibrium 

2.  Constraints      


PACK 
121 
I32 


VII.    THE  PRINCIPLE  OF  VIRTUAL  WORK 143 

VIII.    THEORY  OF  ATTRACTIVE  FORCES: 

1.  Attraction I58 

2.  The  potential 169 

ANSWERS J76 


THEORETICAL   MECHANICS. 

CHAPTER    III. 

INTRODUCTION   TO    DYNAMICS. 

I.   Mass ;  Moments  of  Mass ;  Centroids. 

I.     MASS  J    DENSITY. 

1.  In  the  first  part  of  this  work  only  the  geometrical  and 
Idnematical    properties   of   motion    have   been  considered,  the 
moving  object  being  regarded  as  a  mere  point  or  as  ^a  geo- 
metrical configuration.     It  is,  however,  known,  from  observation 
and  experiment,  that  the  motions  of  actual  physical  bodies  are 
not  fully  described  and  determined  by  those  properties  alone. 

Physical  bodies  are  distinguished  from  geometrical  configura- 
tions by  being  possessed  of  mass ;  and  this  property  as  affecting 
their  motion  must  be  taken  into  account  in  dynamics. 

2.  In  physics  the  mass  of  a  body  is  usually  defined  as  the 
.quantity  of  matter  contained  in  the  body.     Postponing  for  the 
present  the  full  discussion  of  the  idea  of  mass  in  its  relation  to 
.acceleration  and  force,  and  of  the  methods  for  comparing  and 
measuring  masses,  it  will   suffice   for  our  present    purpose   to 
think  of  the  mass  of  a  body  as  a  certain  constant  quantity,  inde- 
pendent of  the  body's  position  or  motion  with  respect  to  the 
•earth  or  other  bodies,  as  an  indestructible  something  underlying 
•every  physical  body. 

The  student  must  be  warned  not  to  confound  mass  with 
weight.  The  weight  of  a  body,  as  we  shall  see  later,  is  the 
force  with  which  the  body  is  attracted  by  the  earth ;  it  varies, 

PART   II — I  I 


2  INTRODUCTION   TO   DYNAMICS.  [3. 

therefore,  with  the  distance  of  the  body  from  the  earth's  centre, 
and  would  vanish  completely  if  the  earth  were  suddenly 
annihilated ;  while  the  indestructibility  of  mass  is  the  first 
fundamental  principle  of  chemistry  and  physics. 

3.  To  compare  the  masses  of  different  bodies,  we  may  adopt 
any  given  body  as  a  standard. 

Thus  in  the  F.  P.  S.  system,  the  standard  mass  is  a  certain 
bar  of  platinum  marked  "  P.  S.,  1844,  I  lb.,"  and  preserved  at 
the  Office  of  the  Exchequer,  London,  England.  This  is  called 
the  " imperial  standard  pound  avoirdupois"  ;  any  mass  equal  to 
it  is  a  unit  of  mass  in  this  system. 

In  the  C.  G.  S.  system,  the  standard  of  mass  is  the  "  Kilo- 
gramme des  archives,"  a  bar  of  platinum  kept  in  the  Palais  des 
archives,  in  Paris,  France.  A  mass  equal  to  one-thousandth  of 
this  standard  is  the  unit  of  mass  in  this  system ;  this  unit  is. 
called  \htgram, 

The  numerical  relation  between  the  British  and  metric  units- 
of  mass  is  as  follows  : 

i  lb.    =453.59265  gm. 

i  gm.  =0.002  204  621  2  lb.  =  15.432  grains. 

4.  The  three  units  of  space,  time,  and  mass  are  called  the 
fundamental  units  of  mechanics,  because  with  the  aid  of  these 
three,  the  units  of  all  other  quantities  occurring  in  mechanics 
can  be  expressed.     Thus  we  have  seen  how  the  units  of  velocity 
and  acceleration  are  based  on  those  of  space  and  time,  and  we 
shall  have  many  more  illustrations  in  what  follows.     Any  unit 
that  can  be  expressed  mathematically  by  means  of  one  or  more 
of  the  fundamental  units  is  called  a  derived  unit. 

5.  From  the  mathematical  point  of  view,  mass  appears  in  our 
dynamical  equations  as  a  coefficient,  generally  to  be  regarded  as 
an  absolute,  positive  constant.     It  serves  to  give  different  values 
(different  valency,  or  "  weight  "  in  the  meaning  of  the  theory  of 
least  squares)  to  the  moving  points,  lines,  areas,  volumes,  apart 
from  their  geometrical  extension. 


9-]  MASS. 

j 

6.  Thus,  a  geometrical  point  endowed  with  mass  is  called  a 
material  particle.  We  may  regard  such  a  mass-point,  or  particle, 
as  the  limit  to  which  a  physical  body  approaches  if  its  volume 
be  imagined  to  decrease  indefinitely,  approaching  the  limit  zero, 
while  its  mass  may  remain  a  finite  quantity.  From  the  physical 
point  of  view  a  particle  must  be  regarded  as  much  an  abstraction 
as  a  geometrical  point,  since  every  finite  physical  mass  occupies 
a  finite  space  and  cannot  be  identified  with  a  point.  We  shall 
see,  however,  that  in  dynamics  this  idea  of  the  mass-point, 
or  particle,  is  of  the  greatest  importance  not  only  because 
physical  matter  is  usually  considered  as  made  up  of  an  aggre- 
gation of  such  points  or  centres  possessing  mass  (molecules, 
atoms),  but  principally  because  in  many  cases  the  motion  of  a 
solid  body  can  be  fully  represented  by  the  motion  of  a  certain 
point  in  it,  called  its  centre  of  mass  or  centroid,  the  whole  mass 
being  regarded  as  concentrated  at  this  point. 

7.  It  is  also  customary  in  dynamics  to  speak  of  material 
lines  and  material  surfaces,  which  may  be  regarded  as  the  limits 
of  physical  bodies  in  which  two  dimensions  or  one  dimension 
have  been  reduced  to  zero.      Thus  a  material  line  represents 
the  limit  of  a  wire,  chain,  or  bar,  in  which  two  dimensions  are, 
neglected ;  a  material  surface  can  be  imagined  as  the  limit  of 
a  thin  shell,  or  lamina,  with  one  dimension  reduced  to  zero. 

8.  A  continuous  mass  of  one,  two,  or  three  dimensions,  is 
said  to  be  homogeneous  if  the  masses  contained  in  any  two  equal 
lengths,  areas,  or  volumes  (as  the  case  may  be),  are  equal.     The 
mass  is  then  said   to  be  distributed  uniformly.      In  all  other 
cases  the  mass  is  said  to  be  heterogeneous. 

9.  The  whole  mass  M  of  a  homogeneous  body  divided  by 
the  space  V  it  fills  is  called  the  density  of  the  mass  or  body  ; 
denoting  density  by  p  we  have  therefore 

M 

*FV 

for  homogeneous  bodies. 


4  INTRODUCTION   TO   DYNAMICS.  [10. 

In  a  heterogeneous  body,  this  quotient  is  called  the  average, 
or  mean,  density.  In  this  case  the  density  at  any  point,  or  the 
density  of  any  space-element  dV,  is  defined  as  the  derivative 

dM 


10.  The  unit  of  density  is  the  density  of  a  substance  such 
that  the  unit  of  volume  contains  the  unit  of  mass.  If  the  units 
of  volume  and  mass  are  selected  arbitrarily,  there  need  not  of 
course  necessarily  exist  any  physical  substance  having  unit 
density  exactly.  Thus  in  the  F.P.S.  system,  unit  density  is 
the  density  of  an  ideal  substance  I  pound  of  which  would  just 
fill  a  cubic-  foot.  As  a  cubic  foot  of  water  has  a  mass  of  62  \ 
pounds,  or  1000  ounces,  the  density  of  water  is  62  \  times  the 
unit  density. 

The  specific  density,  or  specific  gravity,  of  a  substance,  is  the 
ratio  of  its  density  to  that  of  water  at  4°  C.  Let  p  be  the 
density,  p'  the  specific  density,  M  the  mass,  V  the  volume  of 
a  homogeneous  mass,  then  in  British  units 


In  the  C.G.S.  system,  the  unit  of  mass  has  been  so  selected 
as  to  make  the  density  of  water  equal  to  I  very  nearly  ;  in  other 
words,  the  unit  mass  (i  gramme)  of  water,  at  the  temperature  of 
4°  C.,  fills  one  cubic  centimetre. 

In  the  metric  system,  then,  there  is  no  difference  between 
density  and  specific  density  or  specific  gravity. 

r 
2.     MOMENTS    AND    CENTRES    OF    MASS. 

11.  The  product  of  a  mass  m,  concentrated  at  a  point  P,  into 
the  distance  of  the  point  P  from  any  given  point,  line,  or  plane, 
is  called  the  moment  of  this  mass  with  respect  to  the  point,  line, 
or  plane. 


I3.j  MOMENTS   AND   CENTRES   OF   MASS.  5 

Thus,  denoting  by  ry  q,  /,  the  distance  of  the  point  P  from 
the  point  O,  the  line  /,  and  the  plane  TT,  respectively,  we  have 
for  the  moments  of  m  with  respect  to  0,  /,  TT,  the  expressions 
mr,  mq,  mp. 

12.  Let  a  system  of  n  points,  or  particles,  Plt  P^  ...  Pn  be 
given;  let  mlt  m2,  ...mn  be  their  masses,  and  plt  pv  ...pn  their 
distances  from  a  given  plane  TT.  Then  we  call  moment  of  the 
system  with  respect  to  the  plane  TT  the  algebraic  sum 


the  distances  plt  pv  ...  pn  being  taken  with  the  same  sign  or 
opposite  signs  according  as  they  lie  on  the  same  side  or  on 
opposite  sides  of  the  plane  TT. 

It  is  always  possible  to  determine  one  and  only  one  distance 

p  such  that  ^mp  =  Mp,  where  M='£m  —  m1-\-m2-\ \-mn  is  the 

total  mass  of  the  system.  If  this  distance  p  should  happen  to 
be  equal  to  zero,  the  moment  of  the  system  would  evidently 
vanish  with  respect  to  the  plane  TT. 

13.  Let  us  now  refer  the  points  P  to  a  rectangular  system 
of  co-ordinates,  and  let  x,  y,  z  be  their  co-ordinates.  Then  we 
have  for  the  moments  of  the  system  with  respect  to  the  co-ordi- 
nate planes  yz,  zx,  xy,  respectively 


The  point  G  whose  co-ordinates  are 


is  called  the  centre  of  mass,  or  the  centroid,  of  the  system. 

The  centroid  is,  therefore,  defined  as  a  point  such  that  if  the 

. 
whole  mass  M  of  the  system  be  concentrated  at  this  point,  its 


6  INTRODUCTION   TO   DYNAMICS.  [14. 

moment  with  respect  to  any  one  of  the  co-ordinate  planes  is  equal 
to  the  moment  of  the  system. 

14.    It  is  easy  to  see  that  this  holds  not  only  for  the  co-ordi- 
nate planes  but  for  any  plane  whatever.     Let 


be  the  equation  of  any  plane  in  the  normal  form  ; 


the  distances  of  the  points  G,  Pv  P2,  ...,  Pn  from  this  plane. 
Then  we  wish  to  prove  that 

Now  =  < 


hence 


=  Mp. 

The  centroid  can  therefore  be  defined  as  a  point  such  that  its 
moment  with  respect  to  any  plane  is  equal  to  that  of  the  whole 
system,  with  respect  to  the  same  plane. 

It  follows  that  the  moment  of  the  system  vanishes  for  any  plane 
passing  through  the  centroid. 

15.  In  the  case  of  a  continuous  mass,  whether  it  be  of  one, 
two,  or  three  dimensions,  the  same  reasoning  will  apply  if  we 
imagine  the  mass  divided  up  into  elements  dM  of  one,  two,  or 
three  infinitesimal  dimensions,  respectively.  The  summations 
indicated  above  by  2  will  then  become  integrations,  so  that  the 
centroid  of  a  continuous  mass  has  the  co-ordinates 


(xdM 

^  ___  ^/ _ .       t 

$dM  '  $dM 


16.]  -MOMENTS   AND   CENTRES   OF   MASS.  7 

According  as  the  mass  is  of  one,  two,  or  three  dimensions,  a 
single,  double,  or  triple  integration  over  the  whole  mass  will  in 
general  be  required  for  the  determination  of  the  moments 
(xdM,  \ydM%  (zdM  of  the  mass  with  respect  to  the  co-ordi- 
nate planes,  as  well  as  of  the  total  mass  (dM=M. 

Thus,  for  a  mass  distributed  along  a  line  or  a  curve  we  have, 
if  ds  be  the  line-element, 

dM=pds; 

for  a  mass  distributed  over  a  surface-area  we  have,  with  dS  as  a 
surface-element, 


finally,    for   a   mass   distributed    throughout   a  volume   whose 
element  is  dV> 


If  the  mass  be  distributed  along  a  straight  line,  the  centroid 
lies  of  course  on  this  line,  and  one  co-ordinate  is  sufficient  to 
determine  the  position  of  the  centroid.  In  the  case  of  a  plane 
area,  the  centroid  lies  in  the  plane  and  two  co-ordinates  deter- 
mine its  position  ;  we  then  speak  of  moments  with  respect  to 
lines,  instead  of  planes. 

16.  If  the  mass  be  homogeneous  (Art.  8),  i.e.  if  the  density  p 
be  constant,  it  will  be  noticed  that  p  cancels  from  the  numerator 
and  denominator  in  the  equations  (2),  and  does  not  enter  into 
the  problem.  Instead  of  speaking  of  a  centre  of  mass,  we  may 
then  speak  of  a  centre  of  arc,  of  area,  of  volume.  The  term 
centroid  is,  however,  to  be  preferred  to  centre,  the  latter  term 
having  a  recognised  geometrical  meaning  different  from  that  of 
the  former. 

The  geometrical  centre  of  a  curve  or  surface  is  a  point  such 
that  any  chord  through  it  is  bisected  by  the  point  ;  there  are 
but  few  curves  and  surfaces  possessing  a  centre. 


8  INTRODUCTION   TO    DYNAMICS.  [17. 

The  centroid  (Art.  14)  is  a  point  such  that,  for  any  plane 
passing  through  it,  the  moment  of  the  system  is  equal  to  zero. 
Such  a  point  exists  for  every  mass,  volume,  area,  or  arc.  The 
centroid  coincides,  of  course,  with  the  centre,  when  such  a 
centre  exists  and  the  distribution  of  mass  is  uniform. 

17.  'As  soon  as  p  is  given  either  as  a  constant  or  as  a  function 
of  the  co-ordinates,  the  problem  of  determining  the  centroid  of 
a  continuous  mass  is  merely  a  problem   in   integration.      To 
simplify   the   integrations,    it   is   of  importance   to   select   the 
element  in  a  convenient  way  conformably  to  the  nature  of  the 
particular  problem. 

Considerations  of  symmetry  and  other  geometrical  properties 
will  frequently  make  it  possible  to  determine  the  centroid  with- 
out resorting  to  integration.  Thus,  in  a  homogeneous  mass, 
any  plane  of  symmetry,  or  any  axis  of  symmetry,  must  contain 
the  centroid,  since  for  such  a  plane  or  line  the  sum  of  the 
moments  is  evidently  zero  (see  Art.  47). 

It  is  to  be  observed  that  the  whole  discussion  is  entirely 
independent  of  the  physical  nature  of  the  masses  m  which 
appear  here  simply  as  numerical  coefficients,  or  "weights," 
attached  to  the  points  (comp.  Art.  5).  Some  of  the  masses 
might  even  be  negative. 

It  will  be  shown  later  that  the  centre  of  gravity,  as  well  as 
the  centre  of  inertia,  of  a  body  coincides  with  its  centroid. 

18.  The  centroid  can  be  defined  without  any  reference  to  a 
co-ordinate  system  as  follows. 

As  in  Art.  12,  let  there  be  given  a  system  of  n  points 
/>!,  P2,  . ..  Pn  (Fig.  i)  whose  masses  are  mv  m2,  ...  mn.  Taking 
an  arbitrary  origin  O  and  putting  OPl  =  rl,  OP2  =  r2, ...  OPn  =  rn) 
we  may  represent  the  moments  m\r^  m2r2,  ...mnrn  of  the 
given  masses  with  respect  to  O  (Art.  11)  by  lengths  (vectors) 
laid  off  on  OPV  OP^  . . .  OPn.  The  moment  of  the  system  can 
then  be  defined  as  the  geometric  sum  of  these  vectors.  It  is- 
therefore  found  by  geometrically  adding  these  vectors ;  i.e.  we 


20.] 


MOMENTS   AND    CENTRES   OF   MASS. 


have  to  lay  off  from  (9,  on  OPV  Opl  =  mlrl\  from /t,  parallel  to 
OP2,  AA  — W2r2>  etc-  '•>  anc^  finely  j°in  O  to  the  end  pn  of  the  poly- 
gon so  formed ;  then  Opn  is  the  geometric  sum,  or  resultant,  of  the 


Fig.    1. 

vectors  m^,  m<>rv  ...mnrn.  Using  square  brackets  to  indicate 
geometric  addition,  we  have  Opn=^\inr\.  A  point  G  taken  on 
the  line  Opn  so  that 

M-OG=Opn  =  ^[mr],  (3). 

where  M=^m,  is  the  centroid  of  the  system. 

19.  It  is  easy  to  see  that  this  definition  of  the  centroid 
agrees  with  the  one  previously  given  (Art.  13).  For,  to  form 
the  geometric  sum,  or  resultant,  of  the  vectors  m^,  m27'2, 
.  .  .  mnrn,  we  may  resolve  each  of  these  vectors  along  three 
rectangular  axes  drawn  through  O.  The  components  of  m^rv  are 
evidently  m^x^  m^y^  m^,  if  x^  y^  z^  are  the  co-ordinates  of  P19. 
since  x^/r^,  yjr^  z-^/r^  are  the  direction  cosines  of  the  line 
We  find  therefore  for  the  components  of  Opn  the  values 
2my,  ^mz  ;  and  hence  for  the  co-ordinates  of  G, 


20.  The  position  of  the  centroid  G  of  a  given  system  of 
masses  is  independent  of  the  point  O  selected  as  origin.  For 
let  another  point  O'  at  the  distance  d  from  O  be  selected  as 


10  INTRODUCTION   TO   DYNAMICS.  [21. 

origin,  and  let  G'  be  the  point  obtained  as  centroid  from  this 
origin,  so  that 


2[mr],  M-O'G'  =  2|W]. 
As  we  have  the  geometric  equation  [r']  =  [d]  +  [>],  we  find 


Hence  subtracting  the  first  equation  and  dividing  by  M, 

[OfG']-[OG]  =  [d],  or  [O'G']  =  [d]  +  [OG]  =  [O'G] 
.so  that  G  and  Gf  coincide. 

It  follows  from  this  consideration  that  a  given  system  has 
only  one  centroid. 

21.  Regarding  again  the  mass  of  the  centroid    as    equal    to 
that  of  the  whole  system,  we  may  now  define  the  centroid  of  a 
system  as  a  point  such  that  its  moment  with  respect  to  any  point 
or  plane  is  equal  to  the  sum  of  the  moments  of  all  the  points 
constituting  the  system;    the    sum    being   understood    to   be   a 
geometric  sum  for  moments  with  respect  to  a  point,  and   an 
algebraic  sum  for  moments  with  respect  to  a  plane. 

Taking  the  centroid  itself  as  origin,  we  have  the  proposition 
that  the  geometric  sum  of  the  moments  of  a  system  with  respect 
to  the  centroid  is  equal  to  zero.  It  has  been  proved  before 
(Art.  14)  that  the  algebraic  sum  of  the  moments  of  a  system 
vanishes  for  any  plane  passing  through  the  centroid. 

22.  In    determining  the    centroid  of   a  given  system  it  will 
often  be  found  convenient  to  break  the  system  up  into  a  number 
of   partial    systems    whose   centroids  are  either  known  or  can 
be  found  more   readily.      The  moment  of  the  whole  system   is 
obviously  equal  to  the  sum  of  the  moments  of  the  partial  systems. 

Thus  let  the  given  mass  M  be  divided  into  k  partial  masses 
Mlt  M9  ...M»  so  that  M=  M^  +  M^  +  ^+M  ;  let  G,  Gv  G2,  ... 


24-]  DETERMINATION   OF   CENTROIDS.  Ir 

Gk  be  the  centroids  of  M,  Mv  M%,  ...  Mk,  and/,  /p  /2,  ...pk  their 
distances  from  some  fixed  plane.     Then  we  have 


23.   The  particular  case  of  two  partial  systems  occurs  most 
frequently.     We  then  have  with  reference  to  any  plane 


Letting  the  plane  coincide  successively  with  the  three  co-ordi- 
nate planes,  it  will  be  seen  that  G  must  lie  on  the  line  joining 
Glt  G2.  Now  taking  the  plane  at  right  angles  to  G1  G%  through 


Glf  we  have 


similarly  for  a  plane  through 


whence  —^r  =  — 2==    1    a  ; 

J/2       Ml        M 

i.e.  the  centroid  of  the  whole  system  divides  the  distance  of  the 
centroids  of  the  two  partial  systems  in  the  inverse  ratio  of  their 
masses. 

3.     EXAMPLES    OF    THE    DETERMINATION    OF    CENTROIDS. 

24.  Two  Particles.  The  centroid  G  of  two  particles  of  masses 
m^  m<i  concentrated  at  two  points  Pv  P2  lies  on  the  line  P^P^ 
and  divides  the  distance  P-J?%  in  the  inverse  ratio  of  their 
masses,  i.e.  so  that 


(See  Art.  23.)  These  formulae  hold  even  when  one  of  the 
masses  is  positive  and  the  other  negative,  in  which  case  the 
sense  of  the  segments  must  be  attended  to. 


12  INTRODUCTION   TO   DYNAMICS.  [25. 

25.  Three  Particles.  We  find  first  the  centroid  P1  of  mz  at 
P2  and  ms  at  PB  (Fig.  2)  by  Art.  24  ;  then,  by  the  same  rule, 
the  centroid  G  of  m%-}-mB  at  P'  and  m1  at  Pj.  We  might  have 
begun  with  P3  and  Pv  finding  P"  ;  or  with  P:  and  />2,  finding 

/"".  £  lies  at  the  intersection 
of  the  three  lines  PJ>\  P^P", 
PzP'n,  and  can  therefore  be 
constructed  graphically. 

P'  26.   Four  Particles.     Find  the 

Fig'  2>  centroid  P'  of  wx  at  ^  and  m2 

at  Pgj  also  the  centroid  P"  of  w3  at  P3  and  ^/4  at  P±\  then 
the  centroid  ^  of  m1-\-m2  at  Pr  and  mz-\-m±  at  /*". 

The  four  particles  can  be  arranged  in  groups  of  two  in  three 
different  ways.  There  are  therefore  three  lines,  like  P'P",  on 
each  of  which  G  lies.  Any  two  of  these  are  sufficient  to  con- 
struct G  geometrically. 

27.  The  centroid  of  a  homogeneous  rectilinear  segment  (thin 
rod  or  wire  of  constant  cross-section)  is  evidently  at  its  middle 
point. 

28.  If  the  density  of  a  rectilinear  segment  be  proportional  to  the 
nth  power  of  the  distance  from  one  end,  say  p  =  kxn,  we  have 


r 

-    Jo 


n+i 


where  /  is  the  length  of  the  segment. 

(a)  For  n  =  o,  this  gives  x=^l  which  determines  the  centroid 
of  a  homogeneous  straight  segment  (see  Art.  27). 

(b)  For  n=  I,  we  have  x—\  /.     This  determines  the  distance, 
from  the  vertex,  of  the  centroid  of  a  homogeneous  triangular 
area.     For  such  an  area  can  be  resolved  (Fig.  3)  by  parallels 
to  the  base  into  elements  each  of  which  may  be  regarded  as 


29-] 


DETERMINATION   OF   CENTROIDS. 


a  homogeneous  segment  PQ.  If  we  imagine  the  mass  of  every 
such  element  concentrated  at  its  middle  point,  the  homogeneous 
triangle  is  replaced  by  its  median  CO  in 
which  the  density  is  proportional  to  the  dis- 
tance from  the  vertex  C. 

The  centroid  of  a  homogeneous  triangular 
area  lies    therefore   on  the   median  at   two- 
thirds  of  its  length  from  the  vertex ;  as  this 
holds  for  each  median,  the  intersection  of  the  Al 
three  medians  is  the  centroid  (see  Art.  32). 

(c)  For  n  —  2,  we  have  x—\l.  This  gives  the  position  of  the 
centroid  of  a  homogeneous  pyramid  or  cone,  by  reasoning  pre- 
cisely similar  to  that  used  in  (b). 

Thus,  to  find  the  centroid  of  any  homogeneous  pyramid  or 
cone,  join  the  vertex  to  the  centroid  of  the  area  of  the  base  ; 
the  required  centroid  lies  on  this  line  at 
a  distance  equal  to  J  of  its  length  from 
the  vertex. 

29.  Homogeneous  Circular  Arc  (Fig.  4). 
Let  O  be  the  centre,  r  the  radius  of  the 
circle;  ACB  =  s  the  arc,  C  its  middle 
point.  The  centroid  G  must  lie  on  the 
bisecting  radius  OC,  since  this  being  a 
line  of  symmetry,  the  sum  of  the  mo- 
ments of  the  elements  of  the  arc  is  =o 
with  respect  to  this  line  (Art.  17).  To 
find  the  distance  x=OG,  we  take  mo- 
ments with  respect  to  the  diameter  per- 
With  OC  as  axis  of  x,  we  have 


Fig.  4. 

pendicular  to  OC. 


=  r    -ds = 


ds  cos  COP = 

Hence,  s  •  x  —  r-  c,  if  c  be  the  length  of  the  chord  AB. 

If  the  angle  AOB  =  2aoi  the  arc  ^  were  given,  we  might 


I4  INTRODUCTION   TO   DYNAMICS.  [50. 

obtain  the  result  by  taking  the  angle  COP  =6  as  independent 
variable.     We  have  then 


/^ 

=  I 

J— 


rcos     -r     =  2r  sn  a, 


,                                           sin  a 
whence  x=r- 


T-I  •  i_  •*.*.  2  r  sin  a          c       -,  •  -,  .,, 

This   can   be   written  x=r =?'•-,  which  agrees  with 

2ra  s 

the  expression  found  above. 

30.  The  First  Proposition  of  Pappus  and  Guldinus.    If  an  arc  of 

a  plane  curve  be  made  to  rotate  about  an  axis  situated  in  its 
plane,  it  generates  a  surface  of  revolution  whose  surface-area  is 
5=2  IT  (yds,  where  ds  is  the  element  of  the  curve  and  the  axis 
of  rotation  is  taken  as  axis  of  x.  On  the  other  hand  we  have,  if 
s  be  the  length  of  the  generating  arc  and  y  the  ordinate  of  its 
centroid,  s-y  =  \ yds',  hence 

5=2  TT  •  sy=  2  iry  •  s, 

i.e.  the  surface-area  of  a  solid  of  revolution  is  obtained  by  multi- 
plying the  generating  arc  into  the  path  described  by  its  centroid. 

It  is  easy  to  see  that  this  proposition  holds  even  for  incom- 
plete revolutions.  When  the  generating  arc  cuts  the  axis, 
proper  regard  must  be  had  for  signs  and  sense  of  rotation. 

31.  It  follows  from  symmetry  that  the  centroid  of  a  homo- 
geneous circular  or  elliptic  area  (plate,  lamina)  is  at  the  geomet- 
rical centre  of  figure.     Similarly,  the  centroid  of  a  homogeneous 
parallelogram  is  at  the  intersection  of  its  diagonals. 

In  general,  if  a  homogeneous  plane  figure  have  two  axes  of 
symmetry,  the  centroid  must  be  at  the  intersection  of  these 
lines  since  the  sum  of  the  moments  is  zero  for  each  of  these 
lines. 


33-] 


DETERMINATION   OF   CENTROIDS. 


32.  It  has  been  shown  in  Art.  28  (b)  how  the  centroid  of  a 
homogeneous  triangular  area  ABC  can  be  found. 

Dividing   the   area   into   linear   elements   by   drawing   lines 
parallel  to  one  of  the  sides,  say  AB  (Fig.  3,  p.  13),  it  appears 
that  the  centroid  of  each  element,  such  as  PQ,  lies  at  its  middle 
point.     The  locus  of  these  middle  points  is 
the  median  CO  of  the  triangle ;  on  this  line, 
then,  the  centroid  G  of  the  triangle  must  be 
situated.     Resolving  the  triangle  into  linear 
elements  parallel  to  the  side  BC,  or  to  CA, 
it  follows  in  the  same  way  that  G  must  lie  on 
each  of  the  other  two  medians  of  the  triangle. 
The  intersection  of  these  medians  is  there- 
fore the  centroid  G. 

The  point  G  trisects  each  median  so  that  CG/GC'  =  2.  For 
if  A  A'  (Fig.  5)  is  another  median,  the  triangles  AGC  and  A'GC' 
are  similar,  and  A'C'  =  %AC't  hence  CG=\CG. 

It  follows  from  Art.  25,  that  the  centroid  of  the  homogeneous 
triangular  area  coincides  with  that  of  three  particles  of  equal 
mass  placed  at  the  vertices. 

33.  Homogeneous  Quadrilateral.    The  centroid  is  found  graphi- 
cally by  resolving  the  quadrilateral  into  triangles,  finding  their 
centroids,  and  deducing  from  them  the  centroid  of  the  quadri- 
lateral.    This  process  applies  generally  to  any  polygon  and  can 
be  carried  out  in  various  ways. 

Thus  for  the  quadrilateral  A  BCD  (Fig.  6)  drawing  the 
diagonal  AC  and  determining  the  centroids  of  the  triangles 

ABC  and  ADC,  we  obtain  by  join- 
ing these  centroids  one  line  on 
which  the  required  centroid  of  the 
quadrilateral  must  lie.  Repeating 
the  same  construction  for  the  tri- 
angles obtained  by  drawing  the 
other  diagonal  BD,  we  find  a  second  line  on  which  the  centroid 


!6  INTRODUCTION   TO   DYNAMICS.  [34. 

must  lie.     The  intersection  of  these  lines  gives  the  centroid  of 
the  quadrilateral. 

34.  For  some  purposes  it  is  convenient  to  find  a  system  of 
particles  whose  centroid  shall  be  the  same  as  that  of  a  quadrilat- 
eral. The  problem  is  of  course  indeterminate  and  may  be 
solved  in  various  ways. 

Let  m  be  the  mass  of  the  quadrilateral  ABCD  ;  mlt  m%  the 
masses  of  the  triangles  ABC,  ADC.  By  Art.  32,  each  of  these 
triangles  can  be  replaced  by  three  equal  particles  -J^,  \^n^ 
placed  at  the  vertices.  We  thus  have  at  A,  as  well  as  at  C,  a 
mass  |  (*#!  4-  m^  =  ^m. 

The  masses  ^m1  at  B  and  ^m2  at  D,  whose  sum  is  also 
=  \m,  are  proportional  to  the  areas  of  the  triangles  ABC,  ADC, 
or  to  the  lengths  EB,  ED,  if  E  be  the  intersection  of  the 
diagonals.  Now  these  two  different  masses  at  B  and  D  can  be 
replaced  by  a  system  of  three  masses,  \m  at  B,  \m  at  D,  and 
—  \m  at  E.  For  (i)  the  total  mass  evidently  remains  the  same, 
and  (2)  the  centroids  of  the  two  systems  coincide  as  is  easily 
seen  by  taking  moments  with  respect  to  E. 

Indeed,  the  centroid  G'  of  ^ml  at  B  and  ^mz  at  D  is  deter- 
mined by  the  equation 

(ml  4-  m2)  •  EG'  =  m1  •  EB  —  m^  •  ED  ; 

substituting  for  mlt  m%  their  values  as  found  from  the  relations 
ml  +  m2  =  m,  ml/m<2i  =  EB/ED,  this  reduces  to 

m>EG'  =  m-(EB-ED). 

The  centroid  Gu  of  \m  at  B,\m  at  D,  and  —  \m  at  E  is 
given  by 


Hence  G'  and  G"  coincide. 

The  centroid  of  the  area  of  a  homogeneous  quadrilateral  is 
therefore  the  same  as  that  of  four  equal  particles  placed  at  its 


35-] 


DETERMINATION   OF   CENTROIDS. 


vertices   together   with  a  fifth   particle  of  equal   but   negative 
mass,  placed  at  the  intersection  of  the  diagonals. 

35,  In  the  particular  case  of  a  homogeneous  trapezoid  (Fig.  7), 
it  may  be  noticed  that  the  figure  can  be  divided  into  rectilinear 
elements  by  lines  drawn  parallel  to  the  parallel  sides  of  the 
trapezoid.  Every  such  element  has  its  centroid  at  its  middle 
point  ;  the  locus  of  all  these  points  is  the  so-called  median  ;  and 
the  centroid  G  of  the  trapezoid  must  lie  on  this  median,  i.e.  on 
the  line  joining  the  middle  points  E,  F  of  the  parallel  sides. 

To  find  the  ratio  in  which  G  divides  the  length  EF>  we  use 
again  the  method  of  taking  moments.  We  divide  the  trapezoid 


into  two  triangles  by  the  diagonal  BC  and  remember  that  the 
distance  of  the  centroid  of  a  triangle  from  its  base  is  equal  to 
one-third  of  its  height  ;  then  taking  moments  with  respect  to  the 
two  parallel  sides  AB  =  a,  CD=b,  denoting  the  height  of  the 
trapezoid  by  h,  and  the  distances  of  G  from  a  and  b  by  y 
we  obtain 


Dividing,  we  find 


This  gives  the  following  construction  :  Make  AEf  =  b  on  the 
prolongation  of  a,  and  DF1  =a  on  the  prolongation  of  b,  in  the 
opposite  sense  ;  then  E'F'  will  intersect  EF  in  G. 

PART   II  —  2 


i8 


INTRODUCTION   TO    DYNAMICS. 


[36. 


36.  To  find  the  centroid  of  the  cross-section  of  a  T-iron  (Fig.  8) 
it  is  only  necessary  to  find  its  distance  ~x  from  the  lower  side 
AB  ;  for  it  must  lie  on  the  axis  of  symmetry  CD.  Taking 
moments  with  respect  to  AB  we  obtain  with  the  notation 
indicated  in  the  figure  : 


hence 


If  a,  fi  are  nearly  equal  and  very 
small  in  comparison  with  a,  b>  we 
have  approximately 


TG 


_ 

X  - 


a  +  b 


Fig.  8. 


/37.   The  area  of  a  homogeneous  cir- 
cular sector  (Fig.  4, -p.  13)  of  radius  r 

and  angle  AOB  =  2a  can  be  resolved  into  triangular  elements 
POP^  —  ^dQ,  the  bisecting .  radius  OC  being  taken  as  polar 
axis.  The  centroid  of  such  an  element  lies,  by  Art.  32,  at 
the  distance  -|r  from  the  centre  O.  Regarding  the  mass, 
p-^r^dd,  of  each  element  as  concentrated  at  its  centroid,  the 
sector  is  replaced  by  a  homogeneous  circular  arc  of  radius  ^r 
and  density  ^pr*d9.  By  Art.  29,  the  centroid  of  such  an  arc, 
which  is  the  required  centroid  of  the  sector,  lies  on  the  bisect- 
ing radius  OC  at  the  distance  |r«5S?  from  the  centre  O. 

a 
Hence 

-     9    sin  a 


38.    In  general,  for  areas  bounded  by  curves  we  must  resort  to 
integration,  using  the  general  formulae  of  Art.  15. 

If  the  area  5  be  plane,  we  have  in  rectangular  co-ordinates 


39-]  DETERMINATION   OF   CENTROIDS.  !9 

—     Cx*  Cy*  —      Cx*  Cv* 

M-x—\      I    pxdxdy,      M-y  =  \      \    pydxdy; 

•y^j  *Jy^  */*i  \Jv\ 

and  if  the  mass  be  homogeneous,  i.e.  p  =  const.,  since  then  the 
first  integration  can  at  once  be  effected  : 


* 


or  similar  expressions  for  y  as  independent  variable. 

In  polar  co-ordinates,  tKe  element  of  area  is  rdrdd,  and  we 
have  x=r  cos  6,  y  —  r  sin  6  ;  hence 


cos  OdrdO,     S  *y  =  sm  OdrdB; 

or,  performing  the  first  integration, 


Vcos&/0,      5-7=4- 

3 

It  will  be  noticed  that  these  last  formulae  express  also  that 
the  infinitesimal  sector  |  r^dQ  is  taken  as  element,  the  centroid 
of  this  element  having  the  co-ordinates  f  rcosO, 


39.  As  a  somewhat  more  complicated  example  let  us  consider 
a  circular  disc  of  radius  a,  in  which  the  density  varies  directly 
as  the  distance  from  the  centre  (Fig.  9).  Let  a  circle  described 
upon  a  radius  as  diameter  be  cut  out  of  this  disc  ;  it  is  required 
to  find  the  centroid  of  the  remainder. 

Let  O  be  the  centre  of  the  disc  of  radius  a,  C  that  of  the 
disc  of  radius  \a\  Gl  the  centroid  of  the  latter,  G  the  required 
centroid;  and  put  OG1=^1>  OG=x.  Then  if  Ml  be  the  mass 


20  INTRODUCTION   TO   DYNAMICS.  [39. 

of  the   smaller   disc,  M%   that   of  the   larger,  we   must   have 
— J/   •  x = 


Fig.  9. 

The  equation  of  the  smaller  circle  is  r=#  cos#.  Taking  as 
element  of  the  mass  of  the  smaller  disc  the  mass  contained 
between  two  arcs  of  radii  r  and  r+  dr,  we  have  for  this  element  : 


or  since  p=&r,  r=acos0, 

l  =  2  ka*B  cosW  (cos  6). 


Hence         M^kBd  (cos3(9) 


cos3(9  - 


=  f  ka*  ff  cos3/9^  =  |  ka*  -  f  = 


The  centroid  of  the  element  dfJ/j  lies,  according  to  Art.  29, 
at  the  distance  rsm    from  O.     We  have  therefore 


sin  (9  cosW0  •  r 


40.]  DETERMINATION   OF   CENTROIDS. 

The  mass  of  the  larger  disc  is 


21 


Substituting  these  values  into  the  equation  of  moments  we 
find' 

M^XI  6 

x=  ^  *  *    =— —      —a  =  o.i6i6..a. 
M%^Ml    5(3-^-2) 

40.  Proceeding  to  the  determination  of  the  centroids  of 
curved  surface-areas,  we  begin  with 
the  special  case  of  the  homoge- 
neous area  of  a  surface  of  revolu- 
tion. If  the  axis  of  x  coincide 
with  the  axis  of  revolution  and 
R  =  rsm6  be  the  distance  of  any 
point  P  of  the  surface  from  this 
axis  (Fig.  10),  the  equation  of  the 
surface,  or  of  its  meridian  section, 
is  x=f(R} ;  and  the  element  of 
area  is 


Fig.   10. 


dS  = 


=  R  V 


We  have  therefore  for  the  centroid  of  the  portion  of  the  surface 
contained  between  two  sections  at  right  angles  to  the  axis  and 
two  meridian  planes  (i.e.  planes  through  the  axis)  including  an 
angle  0: 


22  INTRODUCTION   TO   DYNAMICS.  [41. 

Similar  formulae  result  when  x  is  taken  as  independent  vari- 
able instead  of  R.  For  a  complete  surface  of  revolution  (£  =  27r 
so  that  j/=o,  z=o,  as  is  otherwise  evident. 

41.  In  the  case  of  spherical  surfaces,  although  the  preceding 
formulae  can  of  course  be  used,  it  is  often  more  convenient  to 
make  use  of  the  geometrical  property  of  the  sphere  that  any 
spherical  area  is  equal  to  the  area  of  its  projection  on  a  cylinder 
circumscribed  about  the  sphere. 

Thus  the  area  on  the  sphere  contained  between  two  parallel 
planes  is  equal  to  the  area  cut  out  by  the  same  two  planes  from 
the  circumscribed  cylinder  whose  axis  is  perpendicular  to  the 
planes.  The  centroid  of  such  a  spherical  area  is  therefore  on 
the  radius  at  right  angles  to  the  bounding  planes  midway 
between  these  planes. 

42.  The  Second  Proposition  of  Pappus  and  Guldinus  (compare 
Art.  30). 

A  plane  area  5  (Fig.  n)  rotating  about  any  axis  situated  in 

in  its  plane  generates  a  solid 
of  revolution  whose  volume  is 
V=TT^(y£-y?)dx,  if  the  axis 
of  revolution  is  taken  as  axis  of 
x  and  j/  are  the  two  ordi- 


F.     j  j  nates  of  the  curve  bounding  the 

area.     On  the  other  hand,  if  y 

be  the  distance  of  the  centroid  G  of  the  plane  area  from  the 
axis,  we  have 


by  Art.  38.     Combining  these  two  results,  we  find 


i.e.  the  volume  of  a  solid  of  revolution  is  obtained  by  multiplying 
the  generating  area  into  the  path  described  by  its  centroid. 

The  proposition  evidently  holds  even  for  a  partial  revolution. 


44-]  DETERMINATION   OF   CENTROIDS.  23 

43.    To  find  the  centroid  of  a  portion  of  any  curved  surface 

F(xy  y,  z)  =  o,  we  have  only  to  substitute  dM=pdS  in  the 
general  formulae  of  Art.  15,  and  then  express  dS  by  the 
ordinary  methods  of  analytic  geometry. 

Denoting  by  /,  m,  n  the  direction  cosines  of  the  normal 
to  the  surface  at  the  point  (x,  y,  z),  and  putting  for  shortness 
•dF/dx=Fx,  dF/dy  =  Fy,  dF/dz=Fz,  we  have 

jc__  dydz  _  dzdx  _  dxdy 
I          m          n 


Fx~  F~  F 
Hence,  substituting 


F. 

in  the  formulae  of  Art.  15,  we  find 


F. 

where  the  integration  is  to  be  extended  over  the  projection  of 
the  portion  of  surface  under  consideration  on  the  plane  xy. 
The  equation  of  the  curve  bounding  this  projection  must  be 
given  :  it  determines  the  limits  of  integration.  It  is  obvious 
how  the  formula  has  to  be  modified  when  the  projection  of  the 
area  on  either  of  the  other  co-ordinate  planes  be  given. 

The  expressions  for  M-x,  M '•  y,  M-z  differ  from  the  above 
expression  for  M  only  in  containing  the  additional  factor 
jr,  y,  z,  respectively,  under  the  integral  sign. 

44.  If  the  equation  of  the  surface  be  given  in  the  form 
z=f(x,y\  as  is  frequently  the  case,  we  have 

F(x,y,z)=z-f(x,y); 


24  INTRODUCTION   TO   DYNAMICS.  [45. 

hence  with  the  usual  Gaussian  notation 

3*     df         dz      df 
—=/=/,  —=^-=4, 
dx     dx  By      dy 

F*=-j>>  Fy=-g,  Fg=I, 
which  gives  M=\    Jx  V  ^  l  +P*  +  f  dxdy, 


px  V  i  +/2  +  g2  dxdy, 
M-y=  ]      \   *py  V  i  +/2  +  q*  dxdy, 

JVl    J*,. 

M-  z=  ]      )pz^/i  +p*  +  q2  dxdy. 

*/V     */«! 


In   the   case  of    a   homogeneous    spherical    surface 
=.a2,  we  have  /  =  dz/dx=  —x/z,    q—ds/dy=—y/z\    hence 
/2  +  ^2  =  ^,  so  that  the  last  of  the  above  formulae  gives 

5  .  z  =  a§  J  dxdy  =  a  •  Sz, 

where  5  is  the  area  of  the  surface  and  Sz  the  area  of  its  pro- 
jection on  the  plane  xy.  The  formula  shows  that  the  distance  z 
of  the  centroid  of  any  spherical  area  5  from  a  plane  passing 
through  the  centre  is  equal  to  the  radius  a  multiplied  by  the 
ratio  of  the  projection  5,  of  the  area  on  the  plane  to  the  area 
itself. 

45.  We  proceed  to  the  methods  of  finding  the  centroids  of 
volumes  or  solids. 

Considerations  of  symmetry  make  it  clear  that  the  centroid 
of  a  homogeneous  parallelepiped  lies  at  the  intersection  of  its 
diagonals  ;  similarly,  that  of  a  homogeneous  prism  or  cylinder 
coincides  with  the  centroid  of  the  area  of  its  middle  section  (i.e. 
a  plane  section  parallel  to,  and  equally  distant  from,  the  bases). 


UNIVE. 

47.]  DETERMINATION   OF   CENTROIDS.  25, 


46.  For  a  homogeneous   pyramid  or  cone,  we  have  found  in 
Art.  28  (c)  that  the  centroid  lies  on  the  line  joining  the  vertex 
to  the  centroid  of  the  area  of  the  base,  at  a  distance  equal  to  £ 
of  this  line  from  the  base.     This  is,  of  course,  easily  shown 
directly  by  resolving  the  pyramid  or  cone  into  plane  elements 
parallel  to  the  base,  in  a  manner  analogous  to  that  used  for  the 
triangular  area  in  Art.  32. 

47.  It  may,  perhaps,  be  well  to  formally  state  the  principal 
laws  of  symmetry  for  homogeneous  solids,  although  they  present 
themselves  so  naturally  that  they  are  used  almost  instinctively. 
For  however  simple  and  obvious  these  propositions  may  appear, 
the  beginner  may  be  led  into  error  if  he  does  not  use  them 
cautiously.     The  proof  rests  on  the  fundamental  definition  of 
the  centroid  as  a  point  such  that  for  any  plane  through  it  the 
sum  of  the  moments  is  zero. 

(a)  If  the  surface  of  the  soiid  have  a  plane  of  symmetry,  i.e.  a 
plane  such  that  every  line  perpendicular  to  it  intersects  the  sur- 
face in  two  points  equidistant  from  the  plane,  the  centroid  lies 
in  this  plane.     Hence,  the  centroid  of  a  homogeneous  solid  is 
at  once  known  if  its  surface  possesses  three  planes  of  symme- 
try.    If  the  surface  has  two  planes  of  symmetry,  the  centroid 
lies  on  their  line  of  intersection. 

(b)  If  the  surface  have  an  axis  of  symmetry,  i.e.  a  line  such 
that  every  line  perpendicular  to  '\\.  intersects  the  surface  in  two 
points  equidistant  from  the  line,  the  centroid  must  lie  on  this 
axis.     Two  axes  of  symmetry  in  the  same  homogeneous  solid 
determine  its  centroid  by  their  intersection. 

(c)  If  the  surface  have  a  centre,  i.e.  a  point  such  that  every 
line  through  it  intersects  the  surface  in  two  points  equidistant 
from  it,  the  centroid  coincides  with  this  centre. 

(d)  If  the  surface  have  a  diametral  plane,  i.e.  a  plane  bisect- 
ing all  chords  that  are  parallel  to  a  certain  direction,  the  centroid 
lies  in  this  plane. 


26  INTRODUCTION    TO   DYNAMICS.  [48. 

48.  Homogeneous  spherical  solids  can  be  treated  by  a  method 
analogous  to  that  used  for  circular  areas   (see  Art.   37).     Thus 
a  homogeneous  spherical  sector  can  be  resolved  into  infinitesimal 
elements,  each  of  which  is  a  pyramid  whose  vertex  lies  at  the 

centre  of  the  sphere  and  whose  base  is 
an  infinitesimal  element  of  the  spherical 
surface  area  of  the  sector.  Such  an 
element,  regarded  as  a  pyramid  (Art. 
46),  has  its  centroid  at  the  distance  |  a 
from  the  centre,  if  a  be  the  radius  of 
the  sphere.  We  may  regard  its  mass  as 
concentrated  at  its  centroid  and  have 
thus  the  solid  sector  replaced  by  a  homo- 
geneous segment  of  a  spherical  area,  of 

radius  \a.     It  has  been  shown  in  Art.  41  that  the  centroid  of 

such  a  segment  bisects  its  height. 

Let  2  a  be  the  angle  at  the  vertex  of  the  given  sector  (Fig.  1 2) ; 

then  the  height  of  the  segment  of  radius  \  a  is  \a(\—  cos  a)  ; 

hence  the  distance  He  of  the  centroid  of  the  solid  spherical  sector 

from  the  centre  is 

x=\a  cos  a  +  ftf  (i  —cos  «)  =  f  a  (i  +cos«)=f  a  cos2-. 

49.  In  a  homogeneous  solid  of  revolution  the  centroid  lies  on 
the  axis  of  .revolution,  since  this  line  is  an  axis  of  symmetry 
(Art.  47  (£)).     Taking  this  line  as  the  axis  of  x,  the  equation 
of  the  surface  of  the  solid  is  determined  by  that  of  the  curve 
bounding  the  generating  area,  say  y=f(x). 

We  select  as  element  the  circular  or  ring-shaped  plate  of 
thickness  dx  contained  between  two  sections  of  the  solid  at 
right  angles  to  the  axis  of  revolution  (Fig.  n,  p.  22).  The 
centroid  of  each  such  element  lies  on  the  axis,  and  the  volume 
of  the  element  is  ir(y£—y?}dx,  if  ylt  jj/2,  are  the  ordinates  of 
the  curve  corresponding  to  the  same  value  of  x.  ^  ( 

We  have,  therefore, 


SI.]  DETERMINATION   OF   CENTROIDS.  27 


It  is  easy  to  see  how  the  formula  has  to  be  modified  when 
only  one  value  or  more  than  two  values  of  y  correspond  to  a 
given  value  of  x. 

50.  In  the  most  general  case  of  any  solid  whatever  the  for- 
mulae of  Art.  15  assume  different  forms  according  to  the  system 
of  co-ordinates  used.     Thus  for  rectangular  Cartesian  co-ordi- 
nates the  element  of  volume  is  dv  =  dxdydz,  and  we  have  : 

M=  j*  J  J/o  dxdydz,     M-  ~x  =  J  J  J  px  dxdydz, 

M-y  =  J  J  J  py  dxdydz,    M-  z  =  j  j  j  pz  dxdydz. 

51.  In  polar  co-ordinates,  i.e.  for  the  radius   vector   ry    the 
co-latitude  6  and  the  longitude  </>  (Fig.  10,  p.  21),  the  element 
of  volume  is  an  infinitesimal  rectangular  parallelepiped  having 
the  concurrent  edges  dr,  rdd,  r  sin  Qd$  ;  hence 


As  ;r=rcos0,  jj/  =  rsin  0cos  </>,  ^=rsin^sin^>,  the  centroid    is 
determined  by  the  equations  : 


sn 

>x=  |  J  Jpr3  sin  (9  co& 
*  sin2  (9  cos 
3  sin2  ^  sin 


28 


INTRODUCTION   TO    DYNAMICS. 


[52. 


52.    As  an  illustration  let  us  determine  the  centroid  of  the 

volume  OABCD  (Fig.  13), 
bounded  by  the  three  co-ordi- 
nate planes  and  the  warped 
quadrilateral  (hyperbolic 
paraboloid)  ABCD.  The  latter 
is  generated  by  the  line  LM 
gliding  along  AB  and  CD  so 
as  to  remain  parallel  to  the 
plane  yz.  The  data  are  OA  — 
CD=a,  OB=b,  OC=AD  =  c. 
We  take  as  element  an 
infinitesimal  prism  PQ  off 


X 

D 

R  z  Q 

M 

P 

c 

z 

Fig.  13. 


base    dxdz   and   height  y.      From  similar   triangles    we   have 
=  (c-z)/c,  and  RL/b=(a-x)/a\  hence 


_rd—X     C  —  Z 

a          c 

Thus  we  find,  rejecting  the  constants  which  cancel  in  numerator 
and  denominator, 

***x(a-x)dx-(*-- 


*x(a-x)(c-z)dxdz 

.     . 


a-x)(c-z}dxd 


I 

_  »/o 


x(a-x)dx     —-—     g 


a* 


X(a—x)dx      a2  — 
2      2 


y=- 


,a—x  c  —  z  j  j 
b dxdz 


b 

2ac 


Finally,  z  =  \c,  by  analogy  with  x. 


53-J  DETERMINATION    OF   CENTROIDS.  29 

53,   Exercises. 

(1)  Three  beads  of  masses  3,  5,  12,  are  strung  on  a  straight  wire 
ivhose  mass  is  neglected,  the  bead  of  mass  5  being  midway  between 
he  other  two.     Find  the  centroid.     (Take  moments  about  the  middle 
Doint.) 

(2)  Show  that  the  centroid  of  three  equal  particles  placed  at  the 
Trtices  of  a  triangle  is  at  the  intersection  of  the  medians  of  the  triangle. 

(3)  Show  that  the  centroid  of  three  masses  m1}  m2,  ms  situated  at 
he  vertices  of  a  triangle  and  proportional  to  the  opposite  sides,  is  at 
;he  centre  of  the  inscribed  circle. 

(4)  Equal  particles  are  placed  at  five  of  the  six  vertices  of  a  regu- 
ar  hexagon.      Find  the  distance  of  the  centroid  from  the  centre  of 
igure. 

(5)  Find  the  centroid  of  a  homogeneous  triangular  frame. 

(6)  Show  that  the  centroid  of  a  homogeneous  semicircular  wire  lies 

2 

it  the  distance  -  r  from  the  centre,  r  being  the  radius. 

7T 

(7)  Find  the  co-ordinates  of  the  centroid  of  the  arc  of  a  quadrant 
of  a  circle  by  using  the  first  proposition  of  Pappus  (Art.  30). 

(8)  Find  the  centroid  of  a  circular  arc  AB  of  angle  A  OB  =  a, 
whose  density  varies  as  the  length  of  the  arc  measured  from  A. 

Find  the  centroids  of  the  following  homogeneous  arcs  of  curves  : 

(9)  Parabola  f=$ax  from  the  vertex  to  the  end  of  the  latus 
rectum. 

(10)  Cycloid  x=  a  (0  —  sm$),y  =  a  (i  —  cos0),  from  cusp  to  cusp. 

( 1 1 )  Half  the  cardioid  r  =  a  ( i  +  cos  0) . 

(12)  Catenary  y=-(e^  +  e~~c)   between  two   points    equally  distant 
from  the  axis  of  x. 

(13)  Common  helix  :  x  =  r  cos0,  y  =  r  sin  9,  z  =  krQ,  from  0  =  o  to 
0  =  9. 

(14)  The  sides  of  a  right-angled  triangle  are  a  and  b.     Find  the  dis- 
tances of  the  centroid  of  the  triangular  area  from  the  vertices. 

(15)  From  a  square  A  BCD  one  corner  EAF  is  cut  off  so  that 
AE  =  %a,  AF—\a,  a  being  the  side  of  the  square.     Find  the  centroid 
of  the  remaining  area. 


30  INTRODUCTION    TO   DYNAMICS.  [53. 

(16)  In  a  trapezoid  the  parallel  sides  are  a,  b,  the  height  is  h,  and 
one  of  the  non-parallel  sides  is  perpendicular  to  the  parallel  sides  ; 
show  that  the  co-ordinates  of  the  centroid  with  a  as  axis  of  x  and  the 

perpendicular  side  as  axis  of  y  are  *  =  <*  +  «*>  + 


(17)  Find  the  centroid  of  the  cross-section  of  a  bar  formed  by 
placing  four  angle-irons    with   their   edges  together,  two  of  the  irons 
having  the  dimensions  a,  b,  a,  ft,  as  in  Fig.  8,  Art.  36,  while  the  other 
two  have  the  dimension  a  different,  say  a'. 

(18)  Find  the  centroid  of  the  cross-section  of  a  U-  iron,  the  length 
of  the   flanges  being  a=  12  in.,  that  of  the  web  2^  =  8  in.,  and  the 
thickness  8  =  i  in.     Deduce  the  general  formula  for  x,  and  an  approxi- 
mate formula  for  a  small  8,  and  compare  the  numerical  results. 

(19)  In  the  cross-section  of  an  unsymmetrical  double  T,  the  flanges 
are  2^=  12  in.,  2  V  =8  in.;  the  web  is   0=  10  in.;  and  the  thickness 
of  each  of  the  two  channel-irons  forming  the  bar  is  8  =  i  in.  throughout  ; 
find  the  centroid. 

(20)  In  a  T-iron  the  width  of  the  flange  is  b,  its  thickness  a  ;  the 
depth  of  the  web  is  a,  its  thickness  ft.    Find  the  distance  of  the  centroid 
from  the  outer  side  of  the  flange  ;  give  an  approximate  expression  and 
investigate  it  for  a  =  b,  a  =  ft  =  ±a. 

(21)  If  one-fourth  be  cut  away  from  a  triangle  by  a  parallel  to  the 
base,  show  that  in  the  remaining  area  the  centroid  divides  the  median 
in  the  ratio  4:5. 

(22)  Prove  that  the  centroid  of  any   plane   quadrilateral   ABCD 
coincides  with  that  of  the  triangle  ACF,  if  the  point  F  be  constructed 
by  laying  vftBF=DE  on  the  diagonal  BD,  E  being  the  intersection 
of  the  diagonals. 

(23)  The  centroid  of  a  homogeneous  semicircular  area  of  radius  r 

lies  at  the  distance  x  =  —  r  from  the  centre. 
371" 

(24)  The  centroid  of  the  area  of  a  homogeneous  circular  segment 
of  radius  r  subtending  at  the  centre  an  angle  2  a  is  at  the  distance 
-  sin3a  3 

>  if  '  is  the  Ch°rd>  h  itS  dis" 


tance  from  the  centre,  and  s  the  arc 


53-]  DETERMINATION    OF   CENTROIDS.  $l 

(25)  A  painter's  palette  is  formed  by  cutting  a  small  circle  of  radius 
b  out  of  a  circular  disc  of  radius  a,  the  distance  between  the  centres 
being  c.     It  is  required  to  find  the  distance  of  the  centroid  of  the 
remainder  from  the  centre  of  the  larger  circle.     (Routh.) 

(26)  The  arch  constructed  of  brick  over  a  door  is  in  the  form  of  a 
quadrant  of  a  circular  ring.     The   door  is   5   ft.  wide ;   i-J-  lengths  of 
brick  are  used  (say  12  in.).     Find  the  centroid  of  the  arch. 

Find  the  co-ordinates  of  the  centroid  for  the  following  plane  areas  : 

(27)  Area  bounded  by  the  parabola  y*=.  ^axy  the  axis  of  x,  and  the 
ordinate  y. 

(28)  Area  bounded   by   the   curve  jy  =  sin.#   from  x=o  \.QX  =  TT 
and  the  axis  of  x. 

(29)  Quadrant  of  an  ellipse. 

(30)  Elliptic  segment  bounded   by  the  chord  joining   the  ends  of 
the  major  and  minor  axes. 

(31)  Show,  by  Art.  28,  that  the  centroid  of  the  surface  of  a  right 
circular  cone  lies  at  a  distance  from  the  base  equal  to  one-third  of 
the  height. 

(32)  Find  the  centroid  of  the  portion  of  the  surface  of  a  right  cir- 
cular cone  cut  out  by  two  planes  through  the  axis  inclined  at  an  angle  <£. 

(33)  Find  the  centroid  of  the  area  of  the  earth's  surface  contained 
between  the  tropic  of  Cancer  (latitude  =  23°  28')  and  the  arctic  circle 
(polar  distance  =  23°  28'). 

(34)  Regarding   the   earth   as   a   homogeneous   sphere    of    density 
10  =  5.5,  how  mucn  would  its  centroid  be  displaced  by  superimposing 
over  the  area  bounded  by  the  arctic  circle  an  ice-cap  of  a  uniform  thick- 
ness of  10  miles? 

(35)  A  bowl  in  the  form  of  a  hemisphere  is  closed  by  a  circular  lid 
of  a  material  whose  density  is  three  times  that  of  the  bowl.     Find  the 
centroid. 

(36)  Determine  the  centroid  of  a  homogeneous  solid  hemisphere. 

(37)  Find  the   centroid  of  a  frustum  of  a  cone,  the   radii  of  the 
bases  being  r^  r2 ;  the  height  of  the  frustum,  h. 

(38)  Show  that  the  formula  for  the  frustum  of  the  cone  applies  like- 
wise to  the  frustum  of  any  pyramid  of  the  same  height  h  if  rlt  r2  are 
any  two  homologous  linear  dimensions  of  the  two  bases. 


32  INTRODUCTION   TO    DYNAMICS.  [53. 

(39)  Find  the  centroid  of  a  solid  segment  of  a  sphere  of  radius  a, 
the  height  of  the  segment  being  h. 

(40)  Show  that,  both  for  a  triangular   area   and   for  a  tetrahedra' 
volume,  the  distance  of  the  centroid  from  any  plane  is  the  arithmetic 
mean  of  the  distances  of  the  vertices  from  the  same  plane. 

(41)  Find  the  centroid  of  the  paraboloid  of  revolution  of  height 
generated  by  the  complete  revolution  of  the  parabola  y2  =  ^ax  about 

;its  axis. 

(42)  The  area  bounded  by  the  parabola  y2=^ax,  the  axis  of  x, 
and  the  ordinate  y=yif  revolves  about  the  tangent  at  the  vertex.     Find 

-the  centroid  of  the  solid  of  revolution  so  generated. 

(43)  The  same  area  as  in  problem  (42)  revolves  about  the  ordinate  ylf 
Find  the  centroid. 

(44)  Find   the   centroid  of  an  octant  of  an  ellipsoid  xP/ 


(45)  The  equations  of  the  common  cycloid  referred  to   a   cusp  as 
•origin  and  the  base  as  axis  of  x  are  x  =  a  (6—  sin#),  y  =  a(i  —  cos#) 

Find  the  centroid  :  (a)  of  the  arc  of  the  semi-cycloid  (i.e.  from  cusp 
to  vertex)  ;  (b)  of  the  plane  area  included  between  the  semi-cycloid  and 
the  base  ;  (c)  of  the  surface  generated  by  the  revolution  of  the  semi- 
cycloid  about  the  base  ;  (d  )  of  the  volume  generated  in  the  same  case  ; 
(e)  of  the  surface  generated  by  the  revolution  of  the  whole  cycloid 
(from  cusp  to  cusp)  about  its  axis,  i.e.  the  line  through  the  vertex  at 
right  angles  to  the  base  ;  (/)  of  the  volume  so  generated. 

(46)  Find  the  centroid  of  a  solid  hemisphere  whose  density  varies 
.  as  the  nth  power  of  the  distance  from  the  centre. 

(47)  From  out  of  the  right  cone  ABC  a  cone  ABD  is  cut  of  the 
•  same  base  and  axis,  but  of  smaller  height.     Find  the  centroid  of  the 

remaining  solid. 

(48)  A  triangle  ABC,  whose  sides  are  a,  b,  c,  revolves  about  an  axis 
situated  in  its  plane.     Find  the  surface  area  and  volume  of  the  solid  so 
generated,  if/,  ^,  r  are  the  distances  of  A,  B,  C  from  the  axis. 

(49)  "  Water  is  poured  gently  into  a  cylindrical  cup  of  uniform  thick- 
ness and  density.     Prove  that  the  locus  of  the  centre  of  gravity  of  the 
water,  the  cup,  and  its  handle  is  a  hyperbola."     (Routh.) 


54-]  DETERMINATION   OF    CENTROIDS.  33 

(50)  Prove  that  the  volume  of  a  truncated  right  cylinder  (i.e.  a  right 
cylinder  cut  by  a  plane  inclined  at  any  angle  to  its  base)  is  equal  to  the 
product  of  the  area  of  its  base  into  the  height  of  the  truncated  cylinder 
at  the  centroid  of  its  base. 

(51)  Prove  that  the  volume  of  a  doubly  truncated  cylinder  is  equal 
to  the  product  of  the  area  of  the  section  at  right  angles  to  the  axis  into 
the  distance  of  the  centroids  of  the  bases. 

54.  For  the  theory  of  moments  and  centres  of  mass  the  student  is 
referred  to  W.  SCHELL,  Theorie  der  Bewegung  und  der  Kr'dfte,  Leipzig, 
Teubner,  Vol.  I.,  1879,  PP-  81— 100;  E.  J.  ROUTH,  Analytical  statics, 
Cambridge,  University  Press,  Vol.  I.,  1891,  pp.  270-314;  J.  SOMOFF, 
Theoretische  Mechanik,  iibersetzt  von  A.  Ziwet,  Leipzig,  Teubner,  Vol.  II., 
1879,  pp.  1-72.  For  problems  see  in  particular  W.  WALTON,  Problems 
in  illustration  of  the  principles  of  theoretical  mechanics,  Cambridge, 
Deighton,  1876,  pp.  1-45  ;  M.  JULLIEN,  Problemes  de  mecanique  ration- 
nelle,  Paris,  Gauthier-Villars,  Vol.  I.,  1866,  pp.  1-46;  F.  KRAFT,  Prob- 
ieme  der  analytischen  Mechanik,  Stuttgart,  Metzler,  Vol.  I.,  1884,  pp. 
527-617.  Compare,  also,  B.  PRICE,  Infinitesimal  calculus,  Oxford, 
•Clarendon  Press,  Vol.  III.,  1868,  pp.  163-206;  MOIGNO,  Lemons  de 
mecanique  analytique,  Statique,  Paris,  Gauthier-Villars,  1868,  pp.  106— 
206 ;  G.  MINCHIN,  Treatise  on  statics,  Oxford,  Clarendon  Press,  Vol. 
I.,  1884,  pp.  261-305  ;  I.  TODHUNTER,  Analytical  static ~s,  edited  by  J.  D. 
Everett,  London,  Macmillan,  1887,  pp.  115-189  ;  W.WALTON,  Problems 
in  elementary  mechanics,  London,  Bell,  1880,  pp.  56-78;  and  for  geo- 
metrical methods,  the  works  on  graphical  statics. 


PART  ii — 3 


34  INTRODUCTION   TO  DYNAMICS.  [55. 


II.   Momentum;  Farce;  Energy. 

55.  Let  us  consider  a  point  moving  with  constant  accelera- 
tion from  rest  in  a  straight  line.  We  know  from  Kinematics 
(Art.  in)  that  its  motion  is  determined  by  the  equations 

v=jt,     s  =  yfl,     J  *»=;>,  (I) 

where  s  is  the  distance  passed  over  in  the  time  /,  v  the  velocity, 
and  j  the  acceleration  at  the  time  t. 

If,  now,  for  the  single  point  we  substitute  an  Mr-tuple  point, 
i.e.  if  we  endow  our  point  with  the  mass  m,  and  thus  make  it  a 
particle  (see  Art.  6),  the  equations  (i)  must  be  multiplied  by  m> 
and  we  obtain 

(2) 


The  quantities  mv,  mjy  \rniP  occurring  in  these  equations 
have  received  special  names  because  they  correspond  to  certain 
physical  conceptions  of  great  importance. 

56.  The  product  mv  of  the  mass   m.   of  a  particle  into  its 
velocity  v  is  called  the  momentum,  or  the  quantity  of  motion,  of 
the  particle. 

57.  In  observing  the  behaviour  of  a  physical  body  in  motion,  we 
notice  that  the  effect  it  produces  —  for  instance,  when  impinging  on 
another  body,  or  more  generally,  whenever  its.  velocity  is  changed  — 
depends  not  only  on  its  velocity,  but  also  on  its  mass.     Familiar  exam- 
ples are  the  following  :  a  loaded  railroad  car  is  not  so  easily  stopped  as- 
an  empty  one  ;  the  destructive  effect  of  a  cannon-ball  depends  both  on 
its  velocity  and  on  its  mass  ;  the  larger  a  fly-wheel,  the  more  difficult  is 
it  to  give  it  a  certain  velocity  ;  etc. 

It  is  from  experiences  of  this  kind  that  the  physical  idea  of  mass  is 
derived. 

The  fact  that  any  change  of  motion  in  a  physical  body  is  affected  by 
its  mass  is  sometimes  ascribed  to  the  so-called  "inertia"  or  "force  of 
inertia,"  of  matter,  which  means,  however,  nothing  else  but  the  property 
of  possessing  mass.  ^  V 


S9.]  MOMENTUM.  35 

58.  Momentum,  being  by  definition  (Art.  56)  the  product  of 
mass  and  velocity,  has  for  its  dimensions  (see  Kinematics,  Art.  92) 

MV  =  MLT~1. 

The  unit  of  momentum  is  the  momentum  of  the  unit  of  mass 
having  the  unit  of  velocity. 

Thus  in  the  C.G.S.  system  the  unit  of  momentum  is  the 
momentum  of  a  particle  of  I  gramme  moving  with  a  velocity  of 
i  cm.  per  second.  There  is  no  generally  accepted  name  for  this 
unit,  although  the  name  bole  was  proposed  by  the  Committee  of 
the  British  Association. 

In  the  F.P.S.  system,  the  unit  is  the  momentum  of  a  particle 
of  one  pound  mass  moving  with  a  velocity  of  I  ft.  per 
second. 

To  find  the  relations  between  these  two  units,  let  there  be  x 
C.G.S.  units  in  the  F.P.S.  unit ;  then 

gm.  cm.         Ib.  ft.  . 

x  •  5 =  i ; 

sec.  sec. 

Ib.      ft. 
hence  . 


gm.    cm. 
or,  by  Art.  3  and  Kinematics,  Art.  14, 

^=453.59x30.48=13825.3; 

i.e.   i  F.P.S.  unit  of 'momentum   =13825.3  C.G.S.  units,  and 
i  C.G.S.  unit  =0.000072331  F.P.S.  units. 

59.   Exercises. 

(1)  What  is  the  momentum  of  a  cannon-ball  weighing  200  Ibs.  when 
moving  with  a  velocity  of  1500  ft.  per  second? 

(2)  With  what  velocity  must  a  railroad-truck  weighing  3  tons  move 
to  have  the  same  momentum  as  the  cannon-ball  in  Ex.  (i)  ? 

(3)  Determine  the  momentum  of  a  one- ton  ram  after  falling  through 
20  feet. 


36  INTRODUCTION   TO    DYNAMICS.  [60. 

60.  The  product   mj    of  the   mass   m    of  a  particle  into  its 
acceleration  j  is  called  force.     Denoting  it  by  F,  we  may  write 
our  equations  (2)  in  the  form 

p 
mv—Ft,     s=\— fi,    ^mv*=Fs.  (3) 

As  long  as  the  velocity  of  a  particle  of  constant  mass  remains 
constant,  its  momentum  remains  unchanged.  If  the  velocity 
changes  uniformly  from  the  value  v  at  the  time  t  to  v'  at  the 
time  t',  the  corresponding  change  of  momentum  is 

mv'  —  mv — mjt!  —  mjt  =  F  (/'—/);  (4) 

hence      .   ,        p-*"^.  "  (?) 

Here  the  acceleration,  and  hence  the  force,  was  assumed  con- 
stant. If  F  be  variable,  we  have  in  the  limit  when  t'  —  t 

becomes  dtt 

j?    d(mv)         dv  //cx 

...   F=*=m-*  '  (6) 

Instead  of  defining  force  as  the  product  of  mass  and  accelera- 
tion, we  may  therefore  define  it  as  the  rate  of  change  of  momen- 
tum with  the  time. 

61.  Integrating  equation  (6),  we  find 

Fdt  =  mv1  —  mv.  (7) 


The  product  F(t'—  t)  of  a  constant  force  into  the  time  t'  —  t  during 

which  it  acts,  and  in  the  case  of  a  variable  force,  the  time- 

Jtr 
Fdt,  is  called  the  impulse  of  the  force  during  this  time. 

It  appears  from  the  equations  (4)  and  (7)  that  the  impulse 
of  a  force  during  a  given  time  is  equal  to  the  change  of  momen- 
tum during  that  time. 

62.  The  idea  of  force  is  no  doubt  primarily  derived  from  the  sensa- 
tion produced  in  a  person  by  the  exertion  of  his  "  muscular,  force." 


64.]  FORCE.  37 

Like  the  sensations  of  light,  sound,  heat,  etc.,  the  sensation  of  exerting 
force  is  capable,  in  a  rough  way,  of  measurement.^  But  the  physiological 
and  psychological  phenomena  attending  the  exertion  of  muscular  force 
when  analysed  more  carefully  are  very  complicated. 

In  ordinary  language  the  term  "  force  "  is  applied  in  a  great  variety  of 
meanings.  For  scientific  purposes  it  is  of  course  necessary  to  attach  a 
single  definite  meaning  to  it. 

63.  In  physics  it  is  customary  to  speak  of  force  as  producing  or 
generating  velocity,  and  to  define    force  as  the  cause  of  acceleration. 
Thus  observation  shows  that  the  velocity  of  a  falling  body  increases 
during  the  fall  ;  the  cause  of  the  observed  change  in  the  velocity,  i.e. 
of  the  acceleration,  is  called  the  force  of  attraction,  and  is  supposed  to 
be  exerted  by  the  earth.     Again,  a  body  falling  in  the  air,  or  in  some 
other  medium,  is  observed  to  increase  its  velocity  less  rapidly  than 
a  body  falling  in  vacuo  ;  a  force  of  resistance  is  therefore  ascribed  to 
the  medium  as  the  cause  of  this  change.     In  a  similar  way  we  speak 
of  the  expansive  force  of  steam,  of  electric  and  magnetic  forces,  etc., 
because  all  these  agencies  produce  changes  of  velocity. 

Now,  any  change  in  the  velocity  v  of  a  body  of  given  mass  m  implies 
a  change  in  its  momentum  mv  ;  and  it  is  this  change  of  momentum,  or 
rather  the  rate  at  which  the  momentum  changes  with  the  time,  which 
is  of  prime  importance  in  all  the  applications  of  mechanics.  It  is  there- 
fore convenient  to  have  a  special  name  for  this  rate  of  change,  and  that 
is  what  is  called  force. 

It  is,  however,  well  to  remember  that  in  using  this  term  "force,"  it  is  not 
intended  to  assert  anything  as  to  the  objective  reality  or  actual  nature 
of  force  and  matter  in  the  ordinary  acceptation  of  these  terms.  Our 
knowledge  comes  to  us  through  our  sense-impressions,  and  these  would 
all  seem  to  reduce  finally  to  changes  of  motion  and  changes  of  momen- 
tum :  these  alone  we  can  perceive  directly. 

64.  The  definition  of  force  (Art.  60)  as  the  product  of  mass 
and  acceleration  gives  the  dimensions  of  force  as 


The  unit  of  force  is  therefore  the  force  of  a  particle  of  unit 
mass  moving  with  unit  acceleration. 

Hence,  in  the  C.G.S.  system,  it  is  the  force  of  a  particle  of 


38  INTRODUCTION    TO    DYNAMICS.  [65. 

i  -r .mime  moving  with  an  acceleration  of  i  cm.  per  second  per 
second.  This  unit  force  is  called  a  dyne. 

The  definition  is  sometimes  expressed  in  a  slightly  different 
form.*  We  may  say  the  dyne  is  the  force  which,  acting  on  a 
gramme  uniformly  for  one  second,  would  generate  in  it  a  velocity 
of  i  cm.  per  second  ;  or  would  give  it  the  C.G.S.  unit  of  acceler- 
ation ;  or  it  is  the  force  which,  acting  on  any  mass  uniformly  for 
one  second,  would  produce  in  it  the  C.G.S.  unit  of  momentum. 

That  these  various  statements  mean  the  same  thing  follows 
from  the  fundamental  formulae  F=mj,  j—vt,  if  F,  m,  t,  v,j  be 
expressed  in  C.G.S.  units. 

65.  In  the  F.P.S.  system,  the  unit  of  force  is  the  force  of  a 
mass  of  i  Ib.  moving  with  an  acceleration  of  i  ft.  per  second 
per  second.     It  is  called  the  poundal. 

66.  The  dyne  and  the  poundal  are   called  the  absolute,  or 
scientific,  units  of  force. 

To  find  the  relation  between  these  two  units,  let  x  be  the 
number  of  dynes  in  the  poundal ;  then  we  have 


hence,  just  as  in  Art.  58, 

^=13825.3; 

i.e.    i  poundal  =  13825.3  dynes,  and   I  dyne  =0.000072331 
poundals. 

67.  Another  system  of  measuring  force,  the  so-called  gravi- 
tation (or  engineering)  system,  is  in  very  common  use,  and  must 
here  be  explained. 

Among  the  forces  of  nature  the  most  common  is  the  force  of 
gravity,  or  the  weight,  i.e.  the  force  with  which  any  physical 
body  is  attracted  by  the  earth.  As  we  have  convenient  and 

*  J.  D.  EVERETT,  C.G.S.  system  of  units,  1891,  p.  23,  24.      ^ 


69.]  FORCE.  39 

accurate  appliances  for  comparing  the  weights  of  different 
bodies  at  the  same  place,  the  idea  suggests  itself  of  selecting 
as  unit  force  the  weight  of  a  certain  standard  mass. 

In  the  metric  gravitation  system  the  weight  of  a  kilogramme 
has  been  selected  as  unit  force ;  in  the  British  gravitation  sys- 
tem, the  weight  of  a  pound  is  the  unit  force. 

68.  There  are  two  serious  objections  to  the  gravitation  system  of 
measuring  force,  one  of  a  practical  nature,  the  other  theoretical.     The 
former  is  that  the  words  "  kilogramme  "  and  "  pound  "  are  thus  used  in  two 
different  meanings,  sometimes,  and  more  correctly,  as  denoting  a  mass, 
sometimes  as   denoting  a  force.     Wherever  an  ambiguity  might  arise 
from  this  double  use,  the  word  "mass"  or  "weight"  must  be  added. 

The  other  objection  is  more  serious.  The  weight  of  a  body,  and 
hence  the  gravitation  unit  of  force,  is  not  a  constant  quantity ;  it  changes 
from  place  to  place  as  it  depends  on  the  value  of  g,  the  acceleration  of 
gravity. 

For,  the  weight  W  of  any  mass  m  being  the  force  with  which  this 
mass  is  attracted  by  the  earth,  we  have 

W=  mg, 

where  g  is  the  acceleration  produced  by  the  earth's  attraction.  Now  it 
is  known  from  experiment  that  this  acceleration  varies  from  place  to 
place ;  according  to  the  law  of  gravitation,  it  is  inversely  proportional 
to  the  square  of  the  distance  from  the  centre  of  the  earth. 

The  weight  of  a  body  is  therefore  a  meaningless  term  unless  the  place 
be  specified  where  the  body  is  situated,  and  the  value  of  g  at  that  place 
be  given. 

It  is  true,  however,  that  the  value  of  g  for  different  points  on  the 
earth's  surface  varies  but  little,  so  that  for  most  practical  purposes 
the  gravitation  system  is  accurate  enough. 

In  the  equations  of  theoretical  dynamics,  in  particular  in  kinetics,  the 
use  of  absolute  units  is  always  understood.  In  statics,  however,  where 
we  are  mainly  concerned  with  the  ratios  of  forces  and  not  with  their 
absolute  values,  gravitation  units  will  generally  be  used  in  the  present 
work  in  view  of  the  practical  applications. 

69.  The  numerical  relation  between  the  absolute  and  gravita- 
tion measures  of  force  is  expressed  by  the  equations 


4o  INTRODUCTION   TO    DYNAMICS.  [70. 

I  kilogramme  (force)  =  1000^-  dynes, 
I  pound  (force)  =g  poundals, 

where  g  is  about  981  in  metric  units,  and  about  32.2  in  British 
units.  In  most  cases  the  more  convenient  values  980  and  32 
may  be  used. 

70.  Exercises. 

(1)  What  is  the  exact  meaning  of  "a  force  of  10  tons"?     Express 
this  force  in  poundals  and  in  dynes. 

(2)  Reduce  2000000  dynes  to  British  gravitation  measure. 

(3)  Express  a  pressure  of  2  Ibs.  per  square  inch  in  kilogrammes  per 
square  centimetre. 

(4)  Prove  that  a  poundal  is  very  nearly  half  an  ounce,  and  a  dyne  a 
little  over  a  milligramme,  in  gravitation  measure. 

(5)  The  numerical  value  of  a  force  being  TOO  in  (absolute)  F.P.S. 
units,  find  its  value  for  the  yard  as  unit  of  length,  the  ton  as  unit  of 
mass,  and  the  minute  as  unit  of  time  (see  Art.  66). 

71.  The  quantity  Jmv2,  i.e.  half  the  product  of  the  mass  of  a 
particle  into  the  square  of  its  velocity,  is  called  the  kinetic  energy 
of  the  particle. 

Let  us  consider  again  a  particle  of  constant  mass  m  moving 
with  a  constant  acceleration,  and  hence  with  a  constant  force  ; 
let  v  be  the  velocity,  s  the  space  described  at  the  time  /;  v',  s' 
the  corresponding  values  at  the  time  t1.  Then  the  last  of  the 
three  fundamental  equations  (see  Arts.  55  and  60)  gives 


F(sf-s)',  (8) 

hence  "       F=\mv*-\m*  ^  >        fe) 

If  F  be  variable,  we  have  in  the  limit 

n^i=  mv<iv. 
ds  ds 


73.]  ENERGY   AND   WORK.  4I 

Force  can  therefore  be  defined  as  the  rate  at  which  the  kinetic 
energy  cJianges  with  the  space.  (Compare  the  end  of  Art.  60.) 

72.  Integrating  the  last  equation  (10),  we  find 

Jv 
Fds  =  ^mv'2  —  ^mv2.  (n) 

The  product  F  (sr  —  s)  of  a  constant  force  F  into  the  space  s'  —  s 
described  in  the  direction  of  the  force,  and  in  the  case  of  a 
variable  force,  the  space-integral  \  Fds,  is  called  the  work  of 
the  force  for  this  space. 

The  equations  (8)  and  (11)  show  that  the  work  of  a  force  is 
equal  to  the  corresponding  change  of  tJie  kinetic  energy. 

We  have  here  assumed  that  the  force  acts  in  the  direction  of 
motion  of  the  particle.  A  more  general  definition  of  work 
including  the  above  as  a  special  case  will  be  given  later  (Art. 
232  sq.). 

The  ideas  of  energy  and  work  have  attained  the  highest 
importance  in  mechanics  and  mathematical  physics  within  com- 
paratively recent  times.  Their  full  discussion  belongs  to 
Kinetics. 

73.  According  to  their  definitions,  both  momentum  (Art.  56) 
and  force  (Art.  60)  may  be  regarded    mathematically    as   mere 
numerical  multiples  of   velocity  and  acceleration,  respectively. 
They  are  therefore  so-called  vector-quantities ;  i.e.  a  momentum 
as  well  as  a  force  can  be  represented  geometrically  by  a  segment 
of   a   straight    line   of    definite   length,    direction,    and    sense. 
Moreover,  as  they  are  referred  to  a  particular  point,  viz.  to  the 
point  whose  mass  is  m,  the  line  representing  a  momentum  or  a 
force  must  be  drawn  through  this  point  ;  the  line  has  therefore 
not  only  direction,  but  also  position  ;  i.e.  a  momentum  as  well 
as  a  force  is  represented  geometrically  by  a  rotor  (compare  Kine- 
matics, Arts.  57,  68,  291  sq.}. 

It  follows  that  concurrent  forces,  for  instance,  can  be  com- 


42  INTRODUCTION    TO    DYNAMICS.  [74. 

pounded  by  geometrical  addition,  as  will  be  explained  more  fully 
in  Chapter  IV. 

On  the  other  hand,  kinetic  energy  and  work  are  not  vector- 
quantities. 

74.  The  ideas  of  momentum,  force,  energy,  work,  with  the  funda- 
mental equations  connecting  them,  as  given  in  the  preceding  articles, 
form  the  groundwork  of  the  whole  science  of  theoretical  dynamics.     The 
application  of  this  science  to  the  interpretation  of  natural  phenomena 
gives  results  in  exact  agreement  with  observation  and  experiment.     It  is 
therefore  important  to  inquire  what  are  the  physical  assumptions  and 
experimental  data  on  which  this  application  of  dynamics  is  based. 

These  assumptions  were  formulated  with  remarkable  clearness  by 
:Sir  Isaac  Newton  in  his  Philosophic  naturalis  principia  mathematica, 
first  published  in  1687,  and  have  since  been  known  as  Newtan's  laws 
of  motion.  As  these  three  axiomata  sive  leges  mofus,  as  Newton  terms 
them,  are  very  often  referred  to  and,  at  least  by  English  writers  on 
dynamics,  are  usually  laid  down  as  the  foundation  of  the  science,*  they 
are  given  here  in  a  literal  translation  : 

I.  Every  body  persists  in  its  state  of  rest  or  of  uniform  motion  along 
a  straight  line,  except  in  so  far  as  it  is  compelled  by  impressed   (i.e. 
external)  forces  to  change  that  state. 

II.  Change  of  motion  is  proportional  to  the  impressed  moving  force 
and  takes  place  along  the  straight  line  in  which  that  force  acts. 

III.  To  every  action  there    is   an   equal  and  contrary  reaction;  or, 
the  mutual  actions  of  two  bodies  on  one  another  are  always  equal  and 
directed  in  contrary  senses. 

75.  Some  explanation  is  necessary  to  correctly  understand  the  mean- 
ing of  these  laws ;    indeed,  Newton's  laws  should   not    be  studied  by 
themselves.     They  become  intelligible  only  if  taken  in  connection  with 
the  definitions  preceding  them  in  the  Principia,  and  with  the  explana- 
tions and  corollaries  that  Newton  himself  has  appended  to  them. 

The  word  "  body  "  must  be  taken  to  mean  particle  ;  the  word  "  motion  " 
in  the  second  law  means  what  is  now  called  momentum. 

All  three  laws  imply  the  idea  of  force  as  the  cause  of  any  change  of 
.momentum  in  a  particle. 

*  See  the  Syllabus  of  elementary  dynamics,  Part  I.,  London,  Macmillan,  1890. 
p.  13  sq.,  prepared  by  the  Association  for  the  Improvement  of  Geometrical  leaching. 


79-]  THE    LAWS    OF    MOTION. 


43 


76.  With  this  definition  of  force  the  first  law,  at  least  in  the  ordinary 
form  of  statement,  for  a  single  particle,  merely  states  that  where  there  is 
no  cause  there  is  no  eifect.     While  this  law  may  appear  superfluous  to  us, 
it  was  not  so  in  the  time  of  Newton.     Kepler  and  Galilei,  less  than  a 
century  before  Newton,  were  the  first  to  insist  more  or  less  clearly  on 
this  so-called  law  of  inertia,  viz.  that  there  is  no  intrinsic  power  or 
tendency  in  moving  matter  to  come  to  rest  or  to  change  its  motion  in 
any  way. 

77.  The  second  law  gives  as  the  measure  of  a  constant  force  the 
amount  of  momentum  generated  in  a  given  time  (see  Art.  60) ;  it  can 
be  called  the  law  of  force.     If  force  be  defined  as  the  cause  of  any 
change  of  momentum,  the  .second  law  follows  naturally  by  assuming,  as 
is  always  done,  that  the  effect  is  proportional  to  the  cause. 

The  first  two  laws  may  thus  be  regarded  from  the  mathematical  point 
of  view  as  nothing  but  a  definition  of  force  ;  but  they  are  certainly 
meant  to  emphasize  the  physical  fact  that  the  assumed  definition  of 
force  is  not  arbitrary,  but  based  on  the  characteristics  of  motion  as 
observed  in  nature. 

In  the  corollaries  to  his  laws  Newton  shows  how  the  composition  and 
resolution  of  forces  by  the  parallelogram  rule  follows  from  his  definition. 
In  deriving  this  result  he  tacitly  assumes  that  the  action  of  any  force  on 
a  particle  takes  place  independently  of  the  action  of  any  other  forces  that 
may  be  acting  on  the  particle  at  the  same  time,  a  principle  that  would 
seem  to  deserve  explicit  statement.  Some  writers  on  mechanics,  in 
particular  French  authors,  prefer  to  replace  Newton's  second  law  by  this 
principle  of  the  independence  of  the  action  of  forces. 

78.  The  third  law  expresses  the  physical  fact  that  in  nature  all  forces 
occur  in  pairs  of  equal  and  opposite  forces.     In  modern  phraseology, 
two  such  equal  and  opposite   forces  in  the  same  line  are  said  to  consti- 
tute a  stress.     Newton's  third  law  is  therefore  called  the  law  of  stress. 

This  law,  which  was  first  clearly  conceived  in  Newton's  time,  involves 
what  may  be  regarded  as  the  second  fundamental  property  of  matter  or 
mass  (the  first  being  its  indestructibility);  viz.  that  any  two  particles  of 
matter  determine  in  each  other  oppositely  directed  accelerations  along  the 
line  joining  them. 

79.  For  a  more  complete  discussion  of  the  physical  laws  underlying 
the   applications   of  theoretical    mechanics,  the   student  is  referred  to 
THOMSON   and   TAIT,   Natural  philosophy,   London,    Macmillan,    1879, 


44  INTRODUCTION   TO   DYNAMICS.  [79. 

Part  I.,  Chapter  II.,  p.  219  sq.;  E.  MACH,  Die  Mechanik  in  ihrer 
Entwickelung,  Leipzig,  Brockhaus,  1889,  p.  203 ;  K.  PEARSON,  The 
grammar  of  science,  London,  Scott,  1892,  p.  357  sq.;  J.  D.  EVERETT, 
C.G.S.  system  of  units,  London,  Macmillan,  1891,  p.  73;  P.  G.  TAIT, 
article,  "  Mechanics,"  in  the  Encyclopedia  Britannica,  9th  ed. ;  J.  CLERK 
MAXWELL,  Matter  and  Motion,  New  York,  Van  Nostrand,  1878;  P.  G. 
TAIT,  Properties  of  matter,  Edinburgh,  Black,  1885. 


8 1.]  INTRODUCTION.  45 


CHAPTER    IV. 

STATICS. 
I.  Introduction. 

80.  When  a  particle  has  two  equal  and  opposite  accelerations 
y,  —  y,  its  motion  will  not  be  changed.     The  same  result  must 
follow  when  a  particle  is  acted  on  by  two  equal  and  opposite 
forces  F=  mj,  F'  =  —mj.    Their  combined  effect  on  the  particle 
is  nil,  so   that    the  particle,  if   originally  at   rest,  will   remain 
at   rest  ;    if    originally   moving   with    constant    velocity   in    a 
straight  line,  it  will  continue  to  do  so  ;  and  if  originally  moving 
under  the  action  of  any  other  forces  in  any  way  whatever,  the 
introduction  of  the  two  equal  and  opposite  forces  will  have  no 
effect  on  its  motion.  . 

We  say  that  two  equal  and  opposite  forces  acting  on  a  particle 
balance,  or  are  equivalent  to  o,  or  are  in  equilibrium.  If  no 
other  forces  act  on  the  particle,  the  particle  itself  is  said  to  be 
in  equilibrium.  It  must  be  kept  in  mind  that  equilibrium  is  not 
synonymous  with  rest. 

81.  Let  us  next  consider  any  two  forces  Fv  F^  acting  simul- 
taneously on  the  same  particle  P  of  mass  m,  and  let  j\,  j^  be 
the  accelerations  produced  by  these  forces  so  that 


The  resultant  acceleration  of  the  particle   is  found  by  geo- 
metrically adding  the  vectors  j\  ,  j^  ;  let  j  be  their  geometric 

sum.     Then  the  force 

F=mj 


46  STATICS.  [82. 

producing  the  resultant  acceleration  is  called  the  resultant  of  the 
forces  Fv  F% ;  these,  or  any  other  two  or  more  forces  having 
the  same  resultant  Ft  are  called  the  components  of  F. 

82.  In  many  investigations  we  are  not   so   much  concerned 
with  the  actual  accelerations  produced  as  with  the  effects  that 
miglit  be  produced  by  any  particular  force  or  system  of  forces  if 
the  particle  or  body  were  perfectly  free  to  move,  i.e.  not  subject 
to  other  forces  or  restraints. 

We  proceed  to  study  the  composition  and  resolution  of  forces 
from  this  point  of  view,  i.e.  without  reference  to  the  accelera- 
tions produced,  but  with  particular  attention  to  the  conditions 
under  which  the  given  system  of  forces  is  in  equilibrium.  This 
study  forms  the  subject  of  Statics. 

83.  The  geometrical  characteristics  of  a  force  are  (a)  its  line 
of  action,  (b)  its  magnitude  or  intensity,  (c)  its  sense.     Properly 
speaking,  two  forces  should   be  called  equal    only  when   they 
agree  in  these  three  characteristics.     But  it  is  customary  to  call 
two  forces  equal  even  when  they  have  only  equal  magnitude  ; 
we  shall  call  them  geometrically  equal,  when  they  agree  in  all 
three  characteristics. 

84.  A  force  acting  on  a  particle  P  is  said  to  have  its  point 
of  application   at   P,  and   the   line   representing   it   is   usually 
drawn  from  P  as  origin.     But  the  point  of  application  is  not  an 


Fig.   14. 

essential  characteristic  of  the  force  ;  it  may  be  taken  at  any 
point  of  its  line  if  this  line  be  regarded  as  rigid.  Thus  the 
force  F  acting  on  the  particle  P  (Fig.  14)  can  be  transferred, 
without  changing  its  effect,  to  any  point  P'  of  its  line  ;  and  two 
equal  and  opposite  forces  in  the  same  line,  such  as  /<\at  P  and 


86.]  INTRODUCTION.  47 

—  F  at  P',  are  in  equilibrium;  provided  always  that  P  and  P] 
may  be  regarded  as  belonging  to  the  same  rigid  body. 

85.    It  follows  from  Arts.  81  and  84  that  any  two  forces  F1 
F2  whose  lines  intersect,  say  at  O'  (Fig.  1 5),  are  equivalent  to, 


i.e.  can  be  replaced  by,  a  single  force  F  called  their  resultant. 
This  resultant  can  be  found  by  replacing  the  forces  Fv  F%  by 
the  equal  forces  F^,  F2f  at  Of,  and  forming  the  parallelogram 
having  F^,  F2'  as  adjacent  sides.  The  diagonal  F'  through  Of 
is  the  required  resultant ;  it  can  be  replaced  by  any  force  F  of 
equal  length  and  sense  in  the  same  line  with  this  diagonal. 

The  parallelogram  construction  need  not  be  made  at  O' ;  we 
may  select  any  origin  O"  (Fig.  15),  draw  through  it  two  vectors 
FJ't  Fz"  equal  (in  direction,  length,  and  sense)  to  Fv  F2,  find 
the  diagonal  F1'  through  O",  and  transfer  it  to  a  parallel  line 
drawn  through  O1. 

Finally,  it  is  not  necessary  to  draw  the  whole  parallelogram  ; 
we  have  only  to  add  the  vectors  Flt  F2  geometrically  from  any 
origin  Oin  (Fig.  15)  and  transfer  their  sum  F"f  to  the  parallel 
through  O'. 

86.  Conversely,  any  force  may  be  resolved  into  two  com- 
ponents along  any  two  lines  intersecting  the  line  of  the  force 


48  STATICS.  [87. 

at  the  same  point  and  lying  in  the  same  plane  with  it.  These 
•components  are  together  equivalent  to  the  force,  i.e.  they  may 
be  substituted  for  the  force. 

87.    It  follows  from  Art.    85   that  the   resultant   R  of   two 
intersecting  forces  P  and  Q,  including  the  angle  6,  is 


For  two  parallel  forces  or  two  forces  acting  in  the  same  line, 
0=o  or  1  80°,  according  as  they  are  of  equal  or  opposite  sense; 
hence  R  =  P+Q  in  the  former  case,  and  R  =  1?—Q  in  the  latter. 
It  is  also  apparent  that  the  resultant  of  any  number  of  parallel 
forces  or  of  forces  acting  in  the  same  line  is  found  as  the 
algebraic  sum  of  these  forces.  How  the  position  of  the  resultant 
is  found  in  the  case  of  parallel  forces  will  be  shown  later  (Arts. 
104,  1  06). 

88.  By  Art.  86,  to  resolve  a  force  R  (Fig.  16)  into  two  com- 
ponents Py  Q  along  two  lines  making  the  angles  «,  /3  with  the 
line  of  Ry  we  have  only  to  draw  through  the  ends  of  a  vector 


2$=R  lines  2  i,  3  i  making  angles  a,  fi  with  2  3  ;  then  2  I  =P, 
i  3  =  Q.     The  triangle  123  gives  the  relations 

P    =    Q  =       R 
sin/3     sin  a     sin(a  +  /3) 

When  the  components  are  at  right  angles,  we  have  P  =  R  cos  a, 
Q  =  R  since. 

89.    The  projection   of  a  closed  polygon  on  any  line  being 
evidently  zero,  and  the  resultant  being  by  definition  the  geo- 


9i.]  INTRODUCTION.  49 

metric  sum  of  its  components,  it  follows  that  the  projection  of 
the  resultant  on  any  line  equals  the  algebraic  sum  of  the  pro- 
jections of  its  components.  This  proposition  is  sometimes 
expressed  in  the  following  form  :  the  resolved  part  of  the 
resultant  in  any  direction  is  equal  to  the  algebraic  sum  of 
the  resolved  parts  of  the  components.  ^ 

Let  /  be  the  line  on  which  we  project  (Fig.  17),  and  let  (/,  R] 
(/,'P),  (l/Q)  denote  the  angles  it  makes  with  the  resultant  R 
and  the  components  P,  Q,  respectively  ;  then 

R  cos  (/,  (R)=P  cos  (/,  P)  +  Q  cos  (/,  '  Q). 

90.  Varignon's  Theorem.  Multiplying  the  last  equation  by  any 
length  OS=s  taken  through  the  initial  point  O  of  R  and  at 
right  angles  to  /,  we  obtain 

R-scos(t,  R)  =  P-scos(l,  P)  +  Q-scos(t,  Q\ 

or  since  s  cos  (/,  R}  —  r,  scos  (/,  P)  =/, 
s  cos  (/,  Q)  =  q,  where  r,  p,  q  are  the 
perpendiculars  let  fall  from  5,  on 
R,  P,  Q,  respectively, 


In  this  form  the  proposition  is  in- 

dependent of  the  direction  of  the     — 

line  /  and  holds  for  any  point  5  in 

the  plane  of  the  parallelogram.  Fig.  17 

91.  Moment  of  a  Force.  The  product  of  a  force  into  its  per- 
pendicular distance  from  a  point  is  called  the  moment  of  the 
force  about  the  point.  It  is  taken  with  the  positive  or  negative 
sign  according  as  the  force  as  seen  from  the  point  is  directed 
counter-clockwise  or  clockwise. 

The  proposition  of  Art.  90,  Pp  +  Qq  =  Rr,  can  now  be  stated 
in  the  following  form  :  the  algebraic  sum  of  the  moments  of  any 
tivo  intersecting  forces  about  any  point  in  their  plane  is  equal  to 
the  moment  of  their  resultant  about  the  same  point. 

PART   II  —  4 


50  STATICS.  [92, 

92.  The  product  Rr  represents  twice  the  area  of  the  triangle 
having  R  for  its  base  and  6"  for  its  vertex  ;  Pp,  Qq  can  be 
interpreted  similarly.  This  remark  leads  to  another  simple 
proof  of  Varignon's  theorem,  which  may  serve  to  make  its 

meaning  better  understood.     With  the 
notation  of  Fig.  18  we  have 

SOR  =  SOQ  +  SQR  +  QOR, 


or  since    ST+  TU=  SU=p, 
Rr=Qq  +  Pp. 


93.  If  the  point  5  be  taken  on  the  resultant  R,  we  have  r=o, 
hence  Pp=  —  Qq  ;  i.e.  the  sum  of  the  moments  of  two  forces  about 

any  point  on  their  resultant  is  zero. 

• 
i  .*•<••' 

94.  The    forces    of    nature   receive    various    special    names 
according    to    the    circumstances    under    which    they    occur. 
Thus  the  weight  of  a  mass  has  already  been  defined  (Art.  67) 
as  the  force  with  which  the  mass  is  attracted  by  the  mass  of 
the  earth. 

When  a  string  carrying  a  mass  at  one  end  is  suspended  with 
its  other  end  from  a  fixed  point,  it  will  be  stretched,  i.e.  sub- 
jected to  a  certain  tension.  This  means  that  if  the  string.  were 
cut  it  would  require  the  application  of  a  force  along  the  line  of 
the  string  to  keep  the  weight  in  equilibrium.  This  force,  which 
may  thus  serve  to  replace  the  action  of  the  string,  is  called  its 
tension. 

When  the  surfaces  of  two  physical  bodies  A,  B  are  in  con- 
tact, a  pressure  may  exist  between  them  ;  that  is,  if  one  of  the 
bodies,  say  B,  be  removed,  it  may  require  the  introduction  of 
a  force  to  keep  A  in  the  same  state  of  rest  or  motion  that  it 
had  before  the  removal  of  B.  This  force,  which  will  obviously 


95-]  INTRODUCTION.  5! 

act  along  the  common  normal  of  the  surfaces  at  the  point  of 
contact,  is  called  the  resistance  of  B,  and  a  force  equal  and 
opposite  to  it  is  called  the  pressure  exerted  by  A  on  B. 

95.   Exercises. 

(1)  Find  the  resultant  of  two  equal  forces  acting  at  right  angles  to 
each  other. 

(2)  Show  that  the  resultant  R  of  two  equal  forces  P  including  an 
angle  0  is  R=  2/?cos((9/2). 

(3)  If  the  resultant  of  two  equal  forces  /'be  equal  to  P,  what  is  the 
angle  between  the  components  ? 

(4)  Find   the    magnitude    and    direction   of    the    resultant   of  two 
forces  of  100  and  200  Ibs.,  including  an  angle  of  60°. 

(5)  Let  R  be  the  effective  piston  pressure  of  a  steam  engine  and  <£ 
the  angle  between  the  direction  of  motion  of  the  piston  and  the  con- 
necting rod  at  any  moment ;  show  that  the  thrust  in  the  connecting  rod 
is  R  sec  cf>  and   the   pressure   on   the   guide-bars  R  tan  <f>.      For  what 
position  of  the  crank  is  the  pressure  on  the  guides  greatest  ? 

(6)  A  weight  W  is  suspended  from  two  fixed  points  A,  B  by  means 
of  a  string  A  CB,  C  being  the  point  of  the  string  where  the  weight  W 
is  attached.     If  AC,  BC  be.  inclined  to  the  vertical  at  angles  a,  (3,  find 
the  tensions  in  AC,  BC :   (a)  analytically;  (b)  graphically. 

(7)  Resolve  a  force  of  20  Ibs.  into  two  components  making  angles 
of  45°  and  30°  with  the  given  force  :  (a)  analytically ;  (b]  graphically. 

(8)  Find   the  rectangular  components  of  a  force  P  if  one  of  the 
components  is  to  make  an  angle  of  30°  with  P. 

(9)  The   resultant  R,  one    of  the    components  P,    and    the   angle 
between  the    two   components,    0  =  60°,    being   given,    find    the   other 
component  Q. 

(10)  A  particle  is  acted  on  by  two  forces  P,  Q  lying  in  the  same 
vertical  plane  and  inclined  to  the  horizon  at  angles  /,  q.     Find  their 
resultant   in    magnitude  and  direction,  if  ^=527  Ibs.,    (2=2 72  Ibs., 
P=  127°  52',  ^  =  32°  13'. 

(n)   Prove  that  the  moments  of  the  two  components  of  a   force 
about  any  point  on  the  line  of  the  force  are  equal  and  opposite. 


52  STATICS.  [95. 

(12)  Two  forces  acting  on  a  point  are  represented  in  magnitude  and 
direction  by  the  tangent  and  normal  of  a  parabola  passing  through  the 
point.     Find  their  resultant,  and  show  that  it  passes  through  the  focus 
of  the  parabola. 

(13)  The  magnitudes  of  two  forces  acting  on  a  point  are  as  2  to  3. 
If  their  resultant  be  equal  to  their  arithmetic  mean,  what  is  the  angle 
between  the  forces? 

(14)  What  is  the  angle  between  a  force  of  i  ton  and  a  force  of  V3 
tons  if  their  resultant  is  2  tons? 

(15)  A  string  with  equal  weights  £F  attached  to  its  ends  is  hung  over 
two  smooth  pegs  A,  B  fixed  in  a  vertical  wall.     Find  the  pressure  on 
the  pegs  :   (a)  when  the  line  AB  is  horizontal ;  (b)  when  it  is  inclined 
to  the  horizon  at  an  angle  0.     The  weight  of  the  string,  its  extensibility 
and  stiffness,  and  the  friction  on  the  pegs  are  neglected  in  this  problem 
as  well  as  in  those  immediately  following. 

( 1 6)  The  string  being  hung  over  three  pegs  A,  B,  C,  determine  graphi- 
cally the  pressures  on  the  pegs.     Let   the  vertical  line   through  B  lie 
between  the  vertical  lines  drawn  through  A  and   C ;    there  will  be  a 
pressure  on  B  only  if  B  lies  above  the  line  AC.    If  B  lies  below  A  C, 
the  pressure  may  be  distributed  over  the  three  pegs  by  passing  the  string 
around  the  peg  B  from  below. 

(17)  In  Ex.  (15),  for  what  position  of  the  line  AB  are  the  pressures 
equal  ? 

(18)  In  Ex.  ( 1 6),  let  A  C  be  horizontal,  and  let  a,  /?,  y  denote  the 
angles  of  the  triangle  AB  C.     What  are  the  pressures  on  the  pegs? 

(19)  In  Ex.  ( 1 8),  what  must   be  the  position  of  B   to   make   the 
pressures  on  the  three  pegs  equal :  (a)  when  B  lies  above  AC ',  (b}  when 
B  lies  below  AC? 

(20)  If  the  string  with  the  equal  weights  W  attached  to  its  ends  be 
strung  over  any  number  of  pegs,  the  pressures  on  the  pegs  are  readily 
determined,  either  graphically  or  analytically,  in  magnitude  and  direc- 
tion ;  these  pressures  depend  only  on  the  value  of  W  and  on  the  angles 
between  the  successive  sides  of  the  polygon  formed  by  the  string,  but 
not  on  the  distances  between  the  pegs. 

(21)  Suppose  the  string  be  closed,  its  ends  being  fastened  together. 
Let  this  string  be  hung  over  three  pegs  A,  B,  C  forming  an  isosceles 
triangle  in  a  vertical  plane  with  its  base  A  C  horizontal,  and  let  a  weight 


95-]  INTRODUCTION.  53 

W  be  suspended  from  the  lowest  point  D  of  the  string.  If  AC=$  ft, 
AB=BC—  2.5  ft.,  and  the  length  of  the  string  2/=  14  ft,  find  the 
tension  of  the  string  and  the  pressures  on  the  pegs. 

(22)  If,  in  Ex.   (21),  the  triangles  AB C  and  ADC  be  equilateral, 
what  would  be  the  tension  and  the  pressures  on  the  pegs  ? 

(23)  In  Ex.  (21),  the  triangles  ABC  and  ADC  being  isosceles  and 
their  common  base  AC  horizontal,  what  must  be  the  relation  between 
the  angles  2  (3  at  B  and  28  at  D  to  make  the  pressures  on  the  three 
pegs  A,  B,  C  equal?     The  pressures   being  made  equal,  what  angle 
gives  the  least  pressure  ? 

(24)  Show,  both  analytically  and  geometrically,  that  a  force  whose 
components  PI,  P2  make  an  angle  6  can  be  resolved  into  two  rectangular 
components  (Pl  +  P2)  cos  (0/2),   (Pi  —  P2)  sin  (0/2 ). 

(25)  In  the  toggle-joint  press  two  equal  rods  CA,  CB  are  hinged  at 
C ;  a  force  Ft  bisecting  the  angle  2  a  between  the  rods  forces  the  ends 
A,  B  apart.    If  A  be  fixed,  find  the  pressure  exerted  at  B  at  right  angles 
to  F  \iF-  100  Ibs.  and  a=  15°,  35°,  65°,  85°,  90°. 


54  STATICS.  [96. 


II.    Concurrent  Forces. 

96.  Let  there  be  given  any  number  n  of  forces  Fv  F2,  FB,  . . ., 
Fn,  whose  directions  all  pass  through  the  same  point.     By  Art. 
85,  we  can  find  the  resultant  R1  of  Fl  and  F2,  next  the  resultant 
R%  of  Rl  and  F&  then  the  resultant  R%  of  R2  and  F^  and  so  on. 
The  resultant  R  of  Rn_2  and  Fn  is  evidently  equivalent  to  the 
whole  system  Fv  Fv  Fs,  . . .,  FM  and  is  called  its  resultant.     We 
thus  have  the  proposition  that  a  system  consisting  of  any  num- 
ber of  concurrent  forces  is  equivalent  to  a  single  resultant. 

97.  It  may  of  course  happen  that  this  resultant  is  zero.     In 
this  case,  the  system  is  said  to  be  in  equilibrium.     The  condition 
of  equilibrium  of  a  system  of  concurrent  forces  is  therefore  R  =  o. 

98.  In  practice,  the  process  of  finding  the  resultant  indicated 
in  Art.  96  is  inconvenient  when  the  number  of  forces  is  large. 


Fig.  19. 

If  the  forces  are  given  graphically,  by  their  vectors,  we  have 
only  to  add  these  vectors  geometrically  (see  Kinematics,  Art.  46), 
and  this  can  best  be  done  in  a  separate  diagram,  called  the  force 
polygon,  or  stress  diagram.  Thus,  in  Fig.  19,  12  is  drawn  equal 
and  parallel  to  Fv  2  3  equal  and  parallel  to  Fv  3  4  to  F9,  4  5  to 
F±,  5  6  to  F&.  The  closing  line  of  the  force  polygon,  viz,,  i  6  in 


joi.]  CONCURRENT    FORCES.  55 

the  figure,  is  equal  and  parallel  to  the  resultant  R,  which  is 
therefore  obtained  by  drawing  through  the  point  of  intersection 
•of  the  forces  a  line  equal  and  parallel  to  i  6. 

The  graphical  condition  of  equilibrium  consists  in  the  closing" 
of  the  force  polygon,  that  is,  in  the  coincidence  of  its  terminal 
point  (6)  with  its  initial  point  (i). 

99.  Analytically, -a  systematic  solution  is  obtained  by  resolv- 
ing each  force  F  into  three  components  X,  V,  Z,  along  three 
rectangular  axes  passing  through  the  point  of  intersection  of 
the  given  forces.  All  components  lying  in  the  direction  of  the 
same  axis  can  then  be  added  algebraically,  and  the  whole  system 
of  forces  is  found  to  be  equivalent  to  three  rectangular  forces 
2X,  2F,  HZ,  which,  by  the  parallelogram  law,  can  be  combined 
into  a  single  resultant 


2F)2+  (2Z)2 

The  angles  a,  /9,  7  made  by  this  resultant  with  the  axes  are 
given  by  the  relations 

cos  tt_cos/3__cos7_  i 
~~~' 


100.    If  the  forces  all  lie  in  the  same  plane,  only  two  axes  are 
required,  and  we  have 


R  =  V(2.T)a  +  (2  F)2,     tan  **' 
where  6  is  the  angle  between  the  axis  of  X  and  R. 

101.  The  condition  of  equilibrium  (Art.  97)  R  =  o  becomes,  by 
Art.  99,  (E^)2+(^F)2+(^Z)2-o.  As  all  terms  in  the  left- 
hand  member  are  positive,  their  sum  can  vanish  only  when  each 
term  is  =  o.  The  analytical  conditions  of  the  equilibrium  of  any 
number  of  concurrent  forces  are  therefore  : 

=0,    2F=o,    2Z=0. 


56  STATICS.  [102. 

102.  As  the  projection  on  any  line  of   any  closed  polygon, 
even  when  its  sides  do  not  all  lie  in  the  same  plane,  is  equal  to 
o,  it  follows  that  the  proposition  of  Art.  89  holds  for  any  num- 
ber of  concurrent  forces. 

103.  Exercises. 

(1)  Show  that  three  forces  that  are  in  equilibrium  must  lie  in  the 
same  plane  and  pass  through  the  same  point. 

(2)  Six  forces  of  i,  2,  3,  4,  5,  6  Ibs.,  respectively,  act  in  the  same 
plane  on  the  same  point,  making  angles  of  60°  with  each  other.     Find 
their  resultant  in  magnitude  and  direction  :    (a)  graphically  ;  (b)  analyti- 
cally. 


(3)   Let  AB  =  c  (Fig.  20)  be  the  vertical  post,  AC  =  b  the  jib,  of  a 
crane,  the  ends  BC  being  connected  by  a  chain  of  length  a.     If  a 
weight  W  be  suspended  from  C,  find  the  tension 
T  produced   by  it  in  the  chain  and  the  thrust  P" 
in  AC. 


(4)  Let  AC  be  hinged  at  A  (Fig.  20)  so  as  to 
turn  freely  in  a  vertical  plane,  and  let  the  chain 
pass  over  a  pulley  at  C  and  carry  the  weight  W. 
In  what  position  of  A  C  will  there  be  equilibrium? 

Fig.  20. 

(5)  Find  the  resultant  R  of  three    concurrent 

forces  A,  B,  C  lying  in  the  same  plane  and  making  angles  a,  (3,  y  with 
each  other. 

(6)  Prove   that   the    moment    of  the   resultant    of  any   number   of 
concurrent   forces   lying   in    the    same    plane    about   any  point  in  this 
plane  is  equal  to  the  sum  of  the  moments  of  the  forces  about  the  same 
point. 

(7)  By  means  of  Ex.  (6),  express  the  conditions  of  equilibrium  of 
any  number  of  concurrent  forces  in  the  same  plane. 

(8)  When  three  forces  are  in  equilibrium,  show  that  they  are  pro- 
portional and  parallel  to  the  sides  of  a  triangle. 

(9)  When  any  number  of  concurrent  forces  are  in  equilibrium,  show 
that  any  one  of  them  reversed  is  the  resultant  of  all  the  others.  '- 


r 

103.]  CONCURRENT    FORCES.  57 

(10)  A  weightless  rod  AC  (Fig.  21),  hinged  at  one  end  A  so  as  to 
be  free  to  turn  in  a  vertical  plane,  is  held  in  a  horizontal  position  by 
means  of  the  chain  BC.  If  a  weight  W  be  suspended  at  C,  find  the 
thrust  Pin  AC  and  the  tension  T  of  the  chain.  Assume  AC—  8  ft., 
AB  =  6  ft. 

(n)  In  Ex.  (10),  suppose  the  rod  AC,  instead  of  being  hinged  at 
A,  to  be  set  firmly  into  the  wall  in  a  horizontal  position ;  and  let  the 
chain  fastened  at  B  run  at  C  over  a  smooth  pulley  and  carry  the 
weight  W.  Find  the  tension  of  the  chain  and  the 
magnitude  and  direction  of  the  pressure  on  the 
pulley  at  C. 


(12)  In  "tacking  against  the  wind,"  let  £Fbe  the 
force  of  the  wind ;  a,  ft  the  angles  made  by  the  axis  | 
of  the  boat  with   the    direction    in  which    the  wind 

blows,  and  with   the   sail,  respectively.      Determine          Fi£-  **• 

the  force  that  drives  the  boat  forward  and  find  for  what  position  of  the 

sail  it  is  greatest. 

(13)  A  cylinder  of  weight  W  rests  on  two  inclined  planes  whose 
intersection  is  horizontal  and  parallel  to  the  axis  of  the  cylinder.     Find 
the  pressures  on  these  planes. 

(14)  Find  the  tensions  in  the  string  ABCD,  fixed  at  A  and  D,  and 
carrying  equal  weights  Wat  B  and  C,  if  AD=c  is  horizontal,  AB=BC 
=  CD,  and  the  length  of  the  string  is  3  /. 

(15)  One  of  the  vertices  A  of  a  regular  hexagon  is  acted  upon  by 
5  forces  represented  in  magnitude  and  direction  by  the   lines  drawn 
from  A  to  the  other  vertices  of  the  hexagon.     Find  their  resultant. 

(16)  Find  the  resultant  of  three  equal  forces  P  acting  on  a  point, 
the  angle  between  the  first  and  second  as  well  as  that  between  the 
second  and  third  being  45°. 

(17)  A  mass  m  rests  on  a  plane  inclined  to  the  horizon  at  an  angle 
0 ;  it  is  kept  in  equilibrium   (a)  by  a  force  P^  parallel  to  the  plane  ; 
(b}  by  a  horizontal  force  P2  \   (/)  by  a  force  P3  inclined  to  the  horizon 
at  an  angle  0  +  a.     Determine  in  each  case  the  force  P  and  the  pres- 
sure R  on  the  plane. 

(18)  Show  that  the  three  forces  represented  by  the  vectors  OA,  OB, 
QC  are  in  equilibrium  if  O  is  the  centroid  of  the  triangular  area  ABC. 


58  STATICS.  [103. 

(19)  Show  that  the   three  vectors    OA,   OB,    OC  have    the    same 
resultant  as  the  three  vectors    OA',   OB1,   OC,  if  A',  B\   C  are  the 
middle  points  of  the  sides  of  the  triangle  ABC. 

(20)  Show  that  the  resultant  of  the  vectors  OA,  OB,  OC  is  OO',  if 
O  is  the  centre  of  the  circle  circumscribed  to  the  triangle  ABC  and 
O'  the  intersection  of  the  altitudes  of  the  same  triangle. 


1 04.] 


PARALLEL   FORCES. 


59 


III.    Parallel  Forces. 

104.  Resultant  of  Two  Parallel  Forces.  The  graphical  con- 
struction of  the  resultant  (Art.  85)  fails  in  the  case  of  parallel 
forces. 

As  an  expedient,  we  may  resolve  one  of  the  two  given  forces 
into  two  components  and  then  combine  these  successively 
with  the  other  force.  Thus,  resolving  P  (Fig.  22)  into  P'  and 
P"  along  the  lines  I  and  II  respectively,  we  may  compound  P" 
with  Q,  and  their  resultant  (acting  along  III)  with  P'.  The 
resolution  of  P  into  two  arbitrary  components  P',  P"  is  best 
done  in  a  separate  diagram,  the  force  polygon,  by  taking  i  2  equal 
and  parallel  to  P,  and  drawing  from  any  arbitrary  point  O, 


Fig.  22. 

called  the  pole,  Oi,  O2,  which  will  represent  the  components 
Pf,  P"  in  magnitude  and  direction.  Then  drawing  2  3  equal 
and  parallel  to  Q,  we  find  O$  as  the  resultant  of  P"  and  Q. 

The  whole  operation  of  finding  the  resultant  R  of  two  paral- 
lel forces  P,  Q  is  therefore  as  follows.  First  construct  \hzforce 
polygon  by  making  I  2  equal  and  parallel  to  P,  23  equal  and  par- 
allel to  Q ;  13  gives  the  magnitude  and  direction  of  the 
resultant  R.  Then  assume  a  pole  O  and  draw  O  I,  O  2,  O$. 
Now  construct  the  so-called  funicular  polygon  (or  equilibrium 
polygon)  by  drawing  in  the  original  figure  a  line  I  parallel  to  O\ 
intersecting  P  say  in/;  through  p  a  line  II  parallel  to  O2  in- 


6o  STATICS.  [105. 

tersecting  Q  say  in  q  ;  through  q  a  line  III  parallel  to  #3.  The 
intersection  r  of  I  and  III  is  a  point  of  the  resultant  R  which  is 
therefore  obtained  in  position  by  drawing  through  r  a  line  equal 
and  parallel  to  I  3. 

105.  In   Fig.   22  the  two  given  parallel   forces  P,   Q  were 
assumed  of  the  same  sense.     The  construction  applies,  however, 
equally  well  to  the  case  when  they  are  of  opposite  sense.     The 
resultant  R  will  then  be  found  to  lie  not  between  P  and  Q,  but 
outside,  on  the  side  of  the  larger  force.     The  construction  fails 
only  when  the  two  given  forces  are  equal  and  of  opposite  sense, 
a    case    that   will    be    considered    later    (see    Art.     112    and 
Arts.  128-138). 

106.  To  determine  the  position  of  R  analytically,  we  may  find 
the  ratio  in   which   it    divides  the  distance   (perpendicular   or 
oblique)  between  P  and  Q.     Let  s  (Fig.  22)  be  the  point  where 
R  meets  pq.     Then,  since  the  triangles  prs  and  O  i  2,  as  well  as 

the  triangles  qsr  and  O  2  3,  are  similar,  we  have 

x 

gs_O_2}     sq  _O2  . 
sr~~  P  '     sr~~  Q 

hence,  dividing,  <c£  =  M. 

sq     P 

This  means  that  the  resultant  of  two  parallel  forces  divides  their 
distance  in  the  inverse  ratio  of  the  forces.  As  this  proposition 
finds  application  in  the  theory  of  the  lever,  it  is  commonly 
referred  to  as  the  principle  of  the  lever. 

Dropping  perpendiculars  /,  q  from  any  point  of  the  resultant 
R  on  the  components  P,  Q,  the  relation  can  be  expressed 
in  the  form 

Pp=-Qq, 

which  shows  that  Varignon's  proposition  of  moments  (Arts, 
89-93)  applies  to  parallel  forces. 


io8.] 


PARALLEL   FORCES. 


6l 


107.  The  resultant  of  two  parallel  forces  can  also  be  found  by 
the  following  simple  process.  Intersect  the  two  parallel  forces 
Py  Q  by  any  transversal  in/  and  q  (Fig.  23)  and  apply  at  these 
points  along  pq  two  equal  and  opposite  forces  F,  —F;  find  the 
resultant  P'  of  F  and  P  and  the  resultant  Pff  of  -F  and  Q  ; 
these  resultants  P1  and  P"  will  intersect  (unless  P  and  Q  be 
equal  and  opposite)  and  their  resultant  R  can  be  found. 


Fig.  23. 

It  will  be  noticed  that  this  construction  reduces  to  that  given 
in  Art.  104  if  for  /^we  select  the  force  2  O  in  the  force  polygon, 
Fig.  22,  p.  59. 

108.  Resultant  of  Any  Number  of  Parallel  Forces.  The  graphi- 
cal method  of  Art.  104  is  readily  extended  to  the  general  case  of 
any  number  of  parallel  forces  lying  in  the  same  plane.  What- 
ever the  number  of  the  forces,  the  force  polygon  gives  magni- 
tude, direction,  and  sense  of  the  resultant,  which  is  simply  the 
algebraic  sum  of  the  given  forces ;  while  the  funicular  polygon 
(formed  by  the  lines  I,  II,  III,  etc.)  gives  the  position  of  the 
resultant  by  furnishing  one  of  its  points,  viz.  the  intersection  of 
the  first  and  last  sides  of  the  funicular  polygon. 

The  process  will  best  be  understood  from  the  following 
example. 

The  horizontal  beam  AB  (Fig.  24).  resting  freely  on  the  fixed  sup- 
ports A,  B  carries  four  weights  W^  W^  W3,  W±. 

To  determine  the  position  of  the  resultant  and  the  reactions  A,  B  of 


62 


STATICS. 


[108. 


the  supports,  construct  the  force  polygon  by  laying  off  in  succession  on  a 
vertical  line  12  =  Wl9  23  =  W2,  34  =  W^  45  =  /^;  select  any  point  O 
as  pole  and  join  it  to  the  points  i,  2,  3,  4,  5. 

Now  we  may  regard  i  O  and  02  as  components  into  which  W^  has 
been  resolved;  similarly  2  6?  and  (9 3  as  components  of  W.2,  3  O  and 
(9  4  as  components  of  lVZt  and  4  (9  and  6*5  as  components  of  W±. 
This  resolution  of  the  weights  into  components  is  transferred  into  the 


Fig.  24. 

main  figure  by  constructing  the  funicular  polygon  as  follows  :  through 
any  point  A'  on  the  direction  of  the  reaction  A  draw  a  parallel  to  O  i 
and  let  it  meet  Wl  in  I ;  through  I  draw  I  II  parallel  to  6>2  ;  through 

II  draw  II  III  parallel  to   #3;  through  III  draw  III  IV  parallel   to 
6>4  ;  and  through  IV  draw  IV  B1  parallel  to  O$  ;  the  point  J31  being  on 
the  direction  of  the  reaction  B. 

If  now  each  weight  be  regarded  as  resolved  along  the  sides  adjacent 
to  it  in  the  funicular  polygon,  since  the  two  components  falling  into 
I  II  are  equal  and  opposite,  and  also  those  falling  into  II  III  and 

III  IV,  the  system  of  weights  is  reduced  to  the  two  componepts  along 


i  io.]  PARALLEL   FORCES.  63 

A'l  and  IV  B'.  The  intersection  of  these  lines,  i.e.  of  the  first  and  last 
sides  of  the  funicular  polygon,  gives  a  point,  R,  of  the  resultant  of 
W,  W*  W\,  W* 

Moreover,  if  the  component  in  A'l  be  resolved  along  A1  1?  and  the 
vertical  through  A1,  and  similarly  the  component  in  J31  IV  along  jB'A' 
and  the  vertical  through  J31,  the  two  components  along  A'£'  will  be 
equal  and  opposite,  each  being  equal  to  the  parallel  Oo  drawn  to  A'Bl 
in  the  force  polygon.  This  parallel  furnishes,  therefore,  the  magnitudes 
of  the  reactions  ^  =  01,^  =  50. 

109.  Analytically,  the  resultant  of   n   parallel  forces  Fv  F2,.. 
...  Fn,  whether  in  the  same  plane  or  not,  can  be  found  as  follows  : 

The  resultant  of  Fl  and  Fz  is  a  force  Fl-{-F2  situated  in  the 
plane  (Flt  F2),  so  that  Flpl  =  F2^2  (Art.  106),  where  /x,/2  are  the 
(perpendicular  or  oblique)  distances  of  the  resultant  from  F1 
and  Fy  respectively.  This  resultant  Fl-{-F2  can  now  be  com- 
bined with  FB  to  form  a  resultant  F^  F2-\-  F3,  whose  distances 
from  FI  +  FI  and  FB  in  the  plane  determined  by  these  two  forces 
are  as  Fs  is  to  /^-f/^.  This  process  can  be  continued  until  all. 
forces  have  been  combined  ;  the  final  resultant  is 

/5  +  F2-K,..  +Fn. 

Any  number  of  parallel  forces  are,  therefore,  in  general  equiva- 
lent to  a  single  resultant  equal  to  their  algebraic  sum. 

110.  To  find  \hz  position  of  this  resultant,  analytically,  let  the 
points  of  application  of  the  forces  Flt  F2,  ...  Fn  be  (x^  y^  Z-L), 
(xvyv  ^2)'  •••  (x™  y™  ^n)-     Tne  Pomt  °^  application  of  the  result- 
ant F^  +  FZ  of  F^  and  F2  may  be  taken  so  as  to  divide  the  dis- 
tance of  the   points  of  application   of  Fl  and  F2  in  the  ratio 
F2:Fl;  hence,  denoting  its  co-ordinates  by  x\  y'  ,  zf,  we  have 

x-x1     or 


(Fl  -f-  F2)  x1  = 
and  similarly  for  y'  and  zl  . 


64  STATICS.  [in. 

The   force  F1  +  F2   combines  with  F3  to  form  a  resultant 
.FI  +  FZ  +  FP  whose  point  of  application  x" ,  yn ,  z"  is  given  by 

and  similar  expressions  for  yn ,  zn . 

Proceeding  in  this  way,  we  find  for  the  point  of  application 
(x,  ~y,  z)  of  the  resultant  of  all  the  given  forces 


with  corresponding  equations  for  y  and  z.     We  may  write  these 
equations  in  the  form  : 

As  these  expressions  for  x,  y,  z  are  independent  of  the  direc- 
tion of  the  parallel  forces,  it  follows  that  the  same  point  (x,  y,  z) 
would  be  found  if  the  forces  were  all  turned  in  any  way  about 
their  points  of  application,  provided  they  remain  parallel.  The 
point  (x,  y,  z)  is  for  this  reason  called  the  centre  of  the  system 
of  parallel  forces.  It  is  nothing  but  what  in  geometry  is  called 
the  mean  point,  or  mean  centre,  of  the  points  of  application  if 
the  forces  are  regarded  as  coefficients  or  "weights"  (in  the 
meaning  of  the  theory  of  least  squares)  of  these  points. 

111.  As  the  origin  of  co-ordinates  in  the  last  article  is  arbi- 
trary, the  equations  (i)  evidently  express  the  proposition  that 
in  any  system  of  parallel  forces  the  sum  of  their  moments  about 
any  point  is  equal  to  the  moment  of  their  resultant  about  the  same 
point.     In  particular,  the  sum  of  the  moments  about  any  point  on 
the  resultant  is  zero. 

This  proposition  may  be  regarded  as  a  generalisation  of  the 
principle  of  the  lever  referred  to  in  Art.  106.  It  furnishes  thei 
convenient  method  of  "taking  moments"  for  the  purpose  of 
determining  the  position  of  the  resultant. 

112.  Couple  of  Forces.     The  construction  given  in  Art.    104 
for  the  resultant  of  two  parallel  forces  fails  only  when  the  two 


1  14-]  PARALLEL   FORCES.  £5 

given  forces  are  equal  and  of  opposite  sense.  In  this  case,  the 
lines  I  and  III  of  the  funicular  polygon  become  parallel,  so 
that  their  intersection  r  lies  at  infinity.  The  magnitude  of  the 
resultant  is  of  course  =o. 

The  combination  of  two  equal  and  opposite  parallel  forces 
(F,  —  F)  is  called  a  couple.  A  couple  is,  therefore,  properly  speak- 
ing, not  equivalent  to  a  single  force,  although  it  may  be  said  to 
be  equivalent  to  a  force  of  magnitude  o  at  an  infinite  distance. 
The  theory  of  couples  will  be  considered  in  detail  in  Arts.  128— 
138- 

113.  Conditions  of  Equilibrium.  We  have  seen  (Art.  109)  that 
a  system  of  n  parallel  forces  is,  in  general,  equivalent  to  a  single 
force  ;  but,  as  appears  from  the  preceding  article,  it  may  happen 
to  reduce  to  a  couple.  It  follows  that  for  the  equilibrium  of  a 
system  of  parallel  forces  the  condition  R  =  o,  though  always  neces- 
sary, is  not  sufficient. 

Now,  if  the  resultant  R  of  the  n  parallel  forces  Fv  F%,  ...  Fn  be 
=o,  the  resultant  R1  of  the  n—  I  forces  Fv  F2,  ...  Fn_^  cannot  be 
o,  and  its  point  of  application  is  found  (by  Art.  1  10)  from 
x  =  (Ffa  +  F^2-\  -----  h  ^-i-^n-i)  /(F1  +  Fz-\  -----  h  /V-i)  and  similar  ex- 
pressions  for  y  and  z.  The  whole  system  of  parallel  forces  is 
therefore  equivalent  to  the  two  parallel  forces  R'  and  Fn.  Two 
such  forces  can  be  in  equilibrium  only  when  they  lie  in  the 
same  straight  line  ;  i.e.  Fn  must  coincide  with  R'  and  must 
therefore  pass  through  the  point  (x,  y,  z),  which  is  a  point  of  R  '. 

The  additional  condition  of  equilibrium  is,  therefore, 


cosa      osc/3 

where  a,  /3,  7  are  the  angles  made  by  the  direction  of  the  forces 
with  the  axes. 

114.    For  practical  application  it  is  usually  best  to  replace  the 
last  condition   by  taking  moments  about  a  convenient  point. 

PART   II  —  5 


66  STATICS.  [115. 

Thus,  the  analytical  conditions  of  equilibrium  can  be  written  in 
the  form 


Graphically,  to  the  former  corresponds  the  closing  of  the  force- 
polygon,  to  the  latter  the  closing  of  the  funicular  polygon. 

115.  Weight  ;  Centre  of  Gravity.  The  most  important  special 
case  of  parallel  forces  is  that  of  the  force  of  gravity  which  acts 
at  any  given  place  near  the  earth's  surface  in  approximately 
parallel  lines  on  every  particle  of  matter. 

If  g  be  the  acceleration  of  gravity,  the  force  of  gravity  on  a 
particle  of  mass  m  is 

w=mg, 

and  is  called  the  weight  of  the  particle  or  of  the  mass  m. 
For  a  system  of  particles  of  masses  mv  m2,  ...  mn  we  have 


The  resultant  W  of  these  parallel  forces, 


where  Mis  the  mass  of  the  system,  is  called  the  weight  of  the 
system. 

The  centre  of  the  parallel  forces  of  gravity  of  a  system  of 
particles  has,  by  Art.  no,  the  co-ordinates 


2<mg 
or  since  the  constant  g  cancels, 


This  point  is  called  the  centre  of  gravity  of  the  system,  and  is 
evidently  identical  with  the  centre  of  mass,  or  centroid  (see 
Art.  13). 


ii6.J 


PARALLEL   FORCES. 


67 


For  continuous  masses  the  same  formulae  hold,  except  that 
the  summations  become  integrations. 

The  weight  W  of.  a  physical  body  of  mass  M  is  therefore  a 
vertical  force  passing  through  the  centroid  of  its  mass. 

116.   Exercises. 

(1)  A  straight  rod  (lever}  of  length  2/=  5  ft.  has  suspended  from  its 
•ends  masses  of  12  and  27  pounds,  respectively.     Find  the  point  (ful- 
.crum)  on  which  it  balances  in  a  horizontal  position  :    (a)   if  its  own 
weight  be  neglected;   (^)  if  it  is  homogeneous  and  weighs  2.2  pounds 
per  running  foot. 

(2)  A  straight  beam  rests  in  a  horizontal  position  on  two  supports 
A,  B.     The  distance  between  the  supports  (the   span)  is  27=24  ft- 
The  beam  carries  a  weight  of  14  tons  at  a  distance  of  8  ft.  from  A,  and 
.a  weight  of  10  tons  at  16  ft.  from  A.     Find  the  pressures  on  the  sup- 
ports (or  the  reactions  of  the  supports):    (a)  when  the  proper  weight  of 
the  beam  is  neglected ;    (b)  when  the  beam  weighs  \  ton  per  running 
foot ;   (c)  when  the  first  third  of  the  beam  (from  A)  weighs  -J-  ton,  the 
second  i  ton,  the  third  %  ton  per  running  foot. 

(3)  A  homogeneous   circular  plate  weighing  W  pounds  rests  in  a 
horizontal  position  on  three  equidistant  supports  near  its  edge,    (a)  What 
is  the  least  weight  P  that  will  upset  it  when  placed  on  the  plate  ?    (b)  If 
there  be  four  equidistant  supports  near  the  edge,  what  is  the  least  weight 
that  will  upset  the  plate  ? 

(4)  Construct  the  resultant  of  two  parallel  forces  of  opposite  sense  by 
the  graphical  method  of  Arts.  104,  105. 

(5)  Solve  exercises  (i)  and  (2)  by  the  graphical  method. 


84 


23560 


610 


33560 


23560 


13300 


Fig.  25. 

(6)  Find  the  reactions  of  the  supports  of  a  bridge  truss  of  50  ft.  span, 
produced  by  a  freight  locomotive  whose  weight  is  distributed  over  the 
three  pairs  of  driving  wheels  and  the  front  truck,  as  indicated  in  Fig.  25  : 
(a)  when  it  stands  in  the  middle  of  the  span ;  (b)  when  its  front  truck 
stands  over  one  support. 


68  STATICS.  [117. 

(7)  Explain  how  the  centroid  of  a  plane  area  can  be  found  graphi- 
cally by  dividing  the  area  into  narrow  parallel  strips. 

(8)  A  homogeneous  rectangular  plate  is  pivoted  on  a  horizontal  axis 
through  its  centre  so  as  to  turn  freely  in  a  vertical  plane.     If  weights 
W^  Wo_,  Wft  W±  be  suspended  from  its  vertices,  what  is  its  position  of 
equilibrium  ? 

(9)  The  ends  of  a  straight  lever  of  length  /  are  acted  upon  by  two 
forces  FI,  F2  in  the  same  plane  with  it,  but  inclined  to  the  lever  at  angles 
«!,  «2.     Determine  the  position  of  the  fulcrum. 

117.  Funicular  Polygons  and  Catenaries.    The  funicular  polygon 
in    its   original   meaning   represents   the   form    of   equilibrium 
assumed  by  a  string  or  cord  suspended  from  two  fixed  points 
and  acted  upon  by  any  forces  in  the  same  plane.     The  "cord" 
is  supposed  to  be  perfectly  flexible,  inextensible,  inelastic,  and 
without  weight.     When  the  number  of  forces  is  made  infinite, 
the  polygon  becomes  a  continuous  curve  called  a  catenary. 

The  present  discussion  is  confined  to  the  case  when  the 
forces  are  all  vertical  so  that  they  can  be  regarded  as  weights. 

118.  Let  A,  B  (Fig.  26)  be  the  fixed  points,  and  let  there  be 
five  weights,  l¥v    W^,   W%,    W^  W%>  suspended  from  the  points 
I,  II,  III,  IV,  V,  of  the  cord. 

If  the  cord  be  cut  on  both  sides  of  the  point  I  and  the  corre- 
sponding tensions  Tv  T2  be  introduced,  the  point  I  must  be 
in  equilibrium  under  the  action  of  the  three  forces  W^,  Tv  7"2. 
Hence  drawing  a  line  I  2  to  represent  the  weight  W±  and  draw- 
ing through  its  ends  I,  2  parallels  to  AI  and  I  II,  respectively, 
we  have  the  force  polygon  of  the  point  I.  Its  sides  O  I  and  2  O 
represent  in  magnitude,  direction,  and  sense  the  tensions  7\, 
T2 ;  in  other  words,  the  weight  W-^  has  thus  been  resolved  into- 
its  components  along  the  adjacent  sides. 

The  same  can  be  done  at  every  vertex  of  the  polygon 
I II  III  IV  V,  and  all  tensions  can  thus  be  found.  But  as  the 
the  tension  7^  in  I  II  occurs  again  (with  sense  reversed)  in  the 
force  polygon  for  the  point  II,  and  so  on,  the  successive 
force  polygons  can  be  fitted  together,  every  triangle  having  one 


120.] 


PARALLEL   FORCES. 


69 


side  in  common  with  the  nexj  one.  Thus  the  complete  force 
polygon  of  the  whole  cord  is  formed,  as  shown  on  the  right  in 
Fig.  26.  Its  vertical  line  represents  the  successive  weights 
Wl=i2,  ^  =  23,  £F3  =  34,  £F4  =  45,  ^=56,  while  the  lines 


Fig.  26. 

radiating  from  the  point  O,  or  pole,    represent   on   the   same 
scale  the  tensions  in  Alt  I  II,  II  III,  III  IV,  IV  V,  V B. 

119.  The  polygon  AI II  III  IV  VB  is  called  the  funicular  poly- 
gon.    It  will  be  noticed  that  if  we  have  given  the  fixed  points 
A,  B,  the  magnitudes  of  the  weights,  their  horizontal  distances, 
say  from  A,  and  the  directions  of  the  first  and  last  sides  AI, 
VB  (whatever  may  be  the  number  of  the  forces),  the  remaining 
sides  of  the  funicular  polygon  can  be  found  by  laying  off  on  a  verti- 
cal line  the  weights  f/F1=  12,  ^  =  23,  etc.,  in  succession,  drawing 
through  i  a  parallel  to  the  first  side,  through  the  end  of  the  last 
weight  (6  in  Fig.  26)  a  parallel  to  the  last  side,  and  joining  the 
intersection  O  of  these  parallels  to  the  points  2,  3,  etc.     The 
sides  of   the  funicular  polygon  must   be   parallel  to  the   lines 
radiating  from  O ;  at  the  same  time  these  lines  represent  the 
tensions  in  these  sides. 

120.  For  the  analytical  investigation,  let  Pt  be  that  vertex  of  a 
funicular  polygon  of  any  number  of  sides  at  which  the  z'th  and 


STATICS. 


[121.. 


i+l 


(i+  i)th  sides  intersect ;  let  «4,  a.+l  be  the  angles  at  which  these 
sides  are  inclined  to  the  horizon,  and  Wt  the  weight  suspended 
from  the  vertex  Pt  (Fig.  27). 

Cutting   the  cord  on  both  sides  of  Pif  and  introducing  the 

tensions  Tt  and  7(+1,  the  condi- 
tions of  equilibrium  of  the  point 
Pt  are  found  by  resolving  the 
three  forces  Wit  Tit  Ti+1  horizon- 
tally and  vertically  (Art.  100)  : 

Tt+1  cos  ai+1=  Ti  cos  ait  (i) 

Ti+1  sin  ai+1  =  T(  sin  at  +  Wt.      (2) 

The    former   of    these    equations 
shows  that,  whatever  the  weights 
W  and  the   lengths   and   inclina- 
tions of   the  sides,   the  horizontal 

components  of  the  tensions  T  are  all  equal.     Denoting  this  con- 

,  - 

stant  value  by  //,  we  have 


Fig.  27. 


7*!  cos  «!  =  T2  cos  «2  =  •  •  •  =  Ti  cos  a.i='"  = 


(3) 


Substituting  the  values  of  Tt  and  Ti+l  as  obtained  from  these 
relations,  into  (2),  this  equation  becomes 


W. 

tan  ai+l  =  tan  «,  +  — ' » 
ri 


(4) 


which  shows  that  as  soon  as  all  the  weights  and  the  inclination 
and  tension  of  any  one  side  are  given,  the  inclinations  and  ten- 
sions of  all  the  other  sides  can  be  found. 

121.  Let  us  now  assume  that  the  weights  W  are  all  equal. 
Then  the  values  of  tan  ai+l  given  by  (4)  form  an  arithmetical 
progression.  If,  in  addition,  we  assume  that  the  sides  of  the 
polygon  are  such  as  to  have  equal  horizontal  projections,  i.e. 
if  we  assume  the  weights  to  be  equally  spaced  horizontally, 


122.]  PARALLEL   FORCES.  7! 

the  vertices  of  the  polygon  will  lie  on  a  parabola  whose  axis 
is  vertical. 

To  find  its  equation,  let  us  suppose,  for  the  sake  of  simplicity, 
that  one  side  of  the  polygon,  say  the  >£th,  is  horizontal  so  that 
ak=o.  Taking  this  side  as  axis  of  x,  its  middle  point  O  as  origin, 
the  co-ordinates  of  the  vertex  Pk  are  ^a,  o,  if  a  be  the  length  of 
the  horizontal  side  and  hence  also  that  of  the  horizontal  projec- 
tion of  every  side. 

Putting  W/H=r,  we  have  tan<*A  =  o,  tan«A+1  =  r,  tan  ak+2 
=  2T,  •••  ;  hence  the  co-ordinates  of  Pk+1  are  x=^a,  y  =  ar  ;  those 
of  Pk+2  are  x  =  ^a,  y  =  aTJr2ar  =  ^ar\  those  of  Pk+3  are  x=*^a, 
=6<2T,  etc.  ;  those  of  the  /zth  vertex  after/^are 


v 


Eliminating  n,  we  find  the  equation 


which  represents  a  parabola  whose  axis  is  the  axis  of  y,  and 
whose  vertex  lies  at  the  distance  \ar  =  \aW/H  below  the 
origin  O. 

122.  Let  the  number  of  sides  be  increased  indefinitely,  the 
length  a  and  the  weight  W  approaching  the  limit  o,  but  so 
that  the  quotient  a/W  remains  finite,  say  lim  (a/  W)=  i/w. 
Then  lim  (a/r}  —  H/w,  lim  (#T)=O;  so  that  the  equation  of  the 
parabola  becomes 


w 

where  w  is  evidently  the  weight  of  the  cord,  or  chain,  per  unit 
length. 

The  parabola  is,  therefore,  the  form  of  equilibrium  of  a  cord 
suspended  from  two  points  when  the  weight  of  the  cord  is  uni- 
formly distributed  over  its  horizontal  projection.  This  is,  for 


STATICS. 


[123. 


instance,  the  case  approximately  in  a  suspension  bridge  with 
uniformly  loaded  roadbed,  the  proper  weight  of  the  chains  being 
neglected. 

123.  This  result  can  easily  be  derived  independently  of 
Art.  121,  by  considering  the  equilibrium  of  any  portion  OP  of 
the  chain  beginning  at  the  lowest  point  O  (Fig.  28).  The  forces 
acting  on  this  portion  are  the  horizontal  tension  H  at  O.  the 
tension  T  along  the  tangent  at  P,  and  the  proper  weight  W  si 
the  chain.  As  this  weight  is  assumed  to  be  uniformly  distribu- 
ted over  the  horizontal  projection  OP' =x  of  OP,  the  weight 
is  W—wx,  and  bisects  OP1. 


Resolving  the  forces  in  the  horizontal  and  vertical  directions, 
we  find,  as  conditions  of  equilibrium, 


ds 


ds 


whence,  eliminating  ds, 


dy  _  w 
dx~  H 


x. 


Integrating  and  considering  that  x=o  when  j=o,  we  find  the 
equation  of  the  parabola  as  above, 

y=^-x* 
y     2H 

124.  The  three  forces  H,  T,  W  are  in  equilibrium  ;  they  must 
intersect  in  a  point  Q  which  bisects  OP',  and  the  force  polygon 
must  be  similar  to  the  triangle  QPP'. 


PARALLEL   FORCES. 


73 


Hence,  if  the  height  of  a  suspension  bridge  be  h,  its  span  2  /, 
its  total  weight  2  W,  we  have  for  the  horizontal  tension  //,  and 
the  tension  T  at  the  point  of  support 

H  T  W 


125.    The  form  of  equilibrium  assumed  by  a  homogeneous  cord 
is  an  ordinary  catenary. 

To  find  its  equation,  we  again  consider  the  equilibrium  of  a 


portion  OP=s  (Fig.  29)  of  the  cord,  beginning  at  the  lowest 
point  O. 

The  weight  of  this  portion  is  now  W=ws,  and  if  a  be  the 
angle  made  by  the  tangent  at  P  with  a  horizontal  line,  we  have 
the  conditions  of  equilibrium 


— 
ds 


ds 


Dividing  and  putting  H/w=c,  we  have  the  differential  equation 

of  the  curve  in  the  form 

dx  _c 
*dy    s 

Substituting  this  value  of  dx]dy  in  the  relation 

we  obtain 

( '  ds^        ,  c2  j  sds 

—  ]=i+  — »    or   dy  =  ± 

^dy)  s2 


74  STATICS.  [126. 


which  gives  by  integration  y+C=  V^-K2,  the  minus  sign  being 
rejected  since  y  increases  with  s. 

The  constant  C  can  be  made  to  disappear  by  taking  the  origin 
Or  on  the  vertical  through  O  at  the  distance  O'O=c  below  the 
lowest  point  O.  We  have,  therefore, 


By  means  of  this  relation,  s  can  be  eliminated  from  the  original 
differential  equation,  and  the  result, 


can  be  integrated  : 

c  log  0  +  Vj^2)  =*+  C. 
=  c  when  x=o,  we  find  C=c\ogc\  hence 


Taking  reciprocals  and  rationalising  the  denominator,  we  find 


hence,  adding  and  subtracting, 


c    *      —*  c 

-(*<  +  *  •)»    *=- 


126.   The  first  equations  of  Art.  125,  Tcosa=ff=ivc,  Tsina 
=  wsy  give  for  the  total  tension  T  at  any  point  P 


Thus,  while  the  horizontal  component  is  constant,  the  vertical 
component  at  any  point  P  is  equal  to  the  weight  of  the  portion  of 
the  cord  from  the  lowest  point  O  to  the  point  P,  and  the  total  ten- 
sion is  equal  to  the  weight  of  a  portion  of  the  cord  equal  to  the 
ordinate  of  the  point  P. 


I27-]  PARALLEL   FORCES.  75. 

Let  Q  be  the  foot  of  the  ordinate  of  P  (Fig.  29),  N  the 
intersection  of  the  normal  with  the  axis  O'x,  and  draw  QR 
perpendicular  to  the  tangent.  Then  PR  =y  sin  a  =  s,  since 
Tsina  =  ws  and  T—wy\  also  QR—ycosa  =  c.  Dividing,  we 
have  tana=s/c;  hence,  differentiating, 

ds        c 


_       _ 


•cos2**  ds     c  da,     cos2  a 

The  figure  shows  that  the  radius  of  curvature  p  is  equal  to  the 
length  of  the  normal  PN. 

The  relation  pco$2a  =  c  shows  further  that  at  the  vertex 
(a  =  o)  the  radius  of  curvature  is  pQ  =  c.  It  follows  that  for  a 
cord  or  chain  suspended  from  two  points  B,  C  in  the  same  hori- 
zontal line,  c  (and  consequently  H)  is  large  when  pQ  is  large,  i.e. 
when  the  curve  is  flat  at  the  vertex;  in  other  words,  when  B  and 
C  are  far  apart. 

127.   Exercises. 

(1)  A  weightless  cord  ABCDEF  is  suspended  from  the  fixed  points 
A,  F,  and  carries  weights  at  the  intermediate  points  J3,  C,  Z>,  2T.    Taking 
A  as  origin,  the  axis  of  x  horizontal,  the  axis  of  y  vertically  upwards,  the 
co-ordinates  of  the  points  B,    C,   D,  £,  F  are  (2,  —  i),  (4,  —1.5), 
(^  —1.5),  (8.5,  —  i),  (10,  2).     If  the  weight  at  B  be  one  pound,  what 
are  the  weights  at  C,  D,  £?    What  are  the  tensions  of  the  sections  of 
the  cord  ?    What  are  the  reactions  of  the  fixed  points  A,  F? 

(2)  The  total  weight  of  a  suspension  bridge  is  2^=50  tons;  the 
span  is  2/=2OO  ft.;    the    height   is   /£=i8ft.     Find  the  tension   of 
the  chain  at  the  ends  and  in  the  middle,  both  graphically  and  analytically. 

(3)  A  uniform  wire  of  length  2  s  is  stretched  between  two  points  in 
the  same  horizontal  line  whose  distance  2x  is  very  nearly  equal  to  2s. 
Find  an  approximate  expression  for  the  parameter  c  of  the  catenary  and 
thence  for  the  tension  of  the  wire. 


STATICS. 


[128. 


IV.    Theory  of  Couples. 

128.  The  combination  of  two  equal  forces  of  opposite  sensed, 
—  F,  acting  along  parallel  lines,  is  called  a  couple  of  forces ;  or 
simply  a  couple  (Art.  112). 

The  perpendicular  distance  AB=p  (Fig.  30)  of  the  forces  of 
the  couple  is  called  the  arm,  and  the  product  Fp  of  the  force  F 
into  the  arm  /  is  called  the  moment  of  the  couple. 

If  we  imagine  the  couple  (F,  p)  to  act  upon  an  invariable 
plane  figure  in  its  plane,  and  if  the  middle  point  of  its  arm  be 

a  fixed  point  of,  this  figure, 
the  couple  will  evidently  tend 
to  turn  the  figure  about  this 
middle  point.  (It  is  to  be 
observed  that  it  is  not  true, 
in  general,  that  a  couple  act- 
ing on  a  rigid  body  produces 
rotation  about  an  axis  at  right 
angles  to  its  plane.)  A  couple 
of  the  type  \F,  p)  or  (F't  p') 
(see  Fig.  30)  will  tend  to  rotate  counter-clockwise,  while  a  couple 
of  the  type  (Fu,pn)  tends  to  turn  clockwise.  Couples  in  the 
same  plane,  or  in  parallel  planes,  are  therefore  distinguished  as 
to  their  sense ;  and  this  sense  is  expressed  by  the  algebraic  sign 
attributed  to  the  moment.  Thus,  the  moment  of  the  couple 
•(F,p)  in  Fig.  30,  is  +  Fp,  that  of  the  couple  (F",prf)  is  -F"p". 

129.  The  effect  of  a  couple  is  not  changed  by  translation. 

Let  AB=p  (Fig.  31)  be  the  arm  of  the  couple  (F,  p)  in  its 
original  position,  and  A'B1  the  same  arm  in  a  new  position  par- 
allel to  the  original  one  in  the  same  plane,  or  in  any  parallel 
plane.  By  introducing  at  each  end  of  the  new  arm  A'Bf  two 
opposite  forces  F,  —F,  each  equal  and  parallel  to  the  original 
forces  F,  the  given  system  is  not  changed  (Art.  80).  'But  the 


Fig.  30. 


13°-] 


THEORY   OF   COUPLES. 


77 


Fig.  31. 


two  equal  and  parallel  forces  F  at  A  and  B1  form  a  resultant 
2F  at  the  middle  point  O  of  the 
diagonal  AB'  of  the  parallelogram 
ABB' A'.  Similarly,  the  two  forces 
—  FatB  and  A'  are  together  equiva-  — | 
lent  to  a  resultant  —2 Fat  the  same 
point  O.  These  two  resultants,  be- 
ing equal  and  opposite  and  acting  in 
the  same  line,  are  together  equiva- 
lent to  o.  Hence  the  whole  system  • 
reduces  to  the  force  F*  at  A'  and  the  force  —  F  at  B\  which 
form,  therefore,  a  couple  equivalent  to  the  original  couple  at 
AB. 

130.  The  effect  of  a  couple  is  not  changed  by  rotation  in  its 
plane. 

Let  AB  (Fig.  32)  be  the  arm  of  the  couple  in  the  original 
position,  C  its  middle  point,  and  let  the  couple  be  turned  about 
C  into  the  position  AB1.  Applying  again  at  A',  B1  equal  and 
opposite  forces  each  equal  to  F,  the  forces  —.Fat  A'  and  Fat  A 
will  form  a  resultant  acting  along  CD,  while  Fat  B1  and  —  F'at 
B  give  an  equal  and  opposite  resultant  along  CE.  These  two 

resultants  destroy  each  other 
and  leave  nothing  but  the 
couple  formed  by  F  at  A'  and 
—  F  at  B',  which  is  therefore 
equivalent  to  the  original 
couple. 

Any  other  displacement  of 
the  couple  in  its  plane,  or  to  a 
parallel  plane,  can  be  effected 
by  a  translation  combined  with 
a  rotation  about  the  middle 
point  of  its  arm  in  its  plane. 
The  effect  of  a  couple  is  therefore  not  changed  by  any  displace- 
ment in  its  plane  or  to  a  parallel  plane. 


73  STATICS.  [131. 

131.  The  effect  of  a  couple  is  not  changed  if  its  force  F  and  its 
arm  p  be  cJianged  simultaneously  in  any  way,  provided  their 
product  Fp  remain  the  same. 

Let  AB=p  be  the  original  arm  (Fig.  33),  F  the  original  force 
of  the  couple ;  and  let  A'£'=J>'  be  the  new  arm.  The  introduc- 
tion of  two  equal  and  opposite  forces  F'  at  A',  and  also  at  B\ 
will  not  change  the  given  system  F,  —F.  Now,  selecting  for 

F'  a  magnitude  such  that  F'p'  =  Fp, 
the  force  F  at  A  and  the  force  —  F1 
F      and  A'   combine  (Arts.   104-106)   to 
A'     form   a  parallel  resultant  through  C, 
the   middle  point  of  the  arm,  since 


B 


--B— i- 


-F 

Fig.  33. 


_F' 


for  this  point  F-  J 

Similarly,  -F   at    B  and   /?'   at   .#' 

give  a  resultant  of  the  same  magni- 

tude,   in   the    same    line    through    C,    but   of   opposite   sense. 

These  two  resultants  thus  destroying  each  other,  there  remains 

only  the  couple  formed  by  F1  at  A'  and  —  F'  at  B\  for  which 

Fp  =  F'p'. 

132.  It  results  from  the  last  three  articles  that  the  only  essen- 
tial characteristics  of  a  couple  are  (a)  the  numerical  value  of  the 
moment  ;  (b)  the  sense,  or  direction  of  rotation  ;  and  (c)  what 
has  been  called  the  "aspect"  of  its  plane,  i.e.  the  direction  of 
any  normal  to  this  plane. 

It  is  to  be  noticed  that  the  plane  of  the  two  forces  forming 
the  couple  is  not  an  essential  characteristic  of  the  couple  ;  just 
as  the  point  of  application  of  a  force  is  not  an  essential  charac- 
teristic of  the  force  (see  Art.  84). 

Now  the  three  characteristics  enumerated  above  can  all  be 
indicated  by  a  vector  which  can  therefore  serve  as  the  geomet- 
rical representative  of  the  couple.  Thus,  the  couple  formed 
by  the  forces  F,  —F  (Fig.  34),  whose  perpendicular  distance 
is  /,  is  represented  by  the  vector  AB  =  Fp  laid  off  on  any 
normal  to  the  plane  of  the  couple.  The  sense  is  indicated  by 


I33-] 


THEORY   OF.  COUPLES. 


79 


drawing  the  vector  toward  that  side  of  the  plane  from  which 
the  couple  is  seen  to  rotate  counter-clockwise. 


Fig.  34. 

We  shall  call  this  geometrical  representative  AB  of  the 
couple  simply  the  vector  of  the  couple.  It  is  sometimes  called 
its  moment,  or  its  axis,  or  its  axial  moment. 

133.  As  was  pointed  out  in  Art.  1 1 2,  a  couple  is  equivalent 
to  a  single  force  acting  along  a  line  at  infinity.  Couples  are, 
therefore,  used  in  statics  to  avoid  the  introduction  of  such 
forces  whose  line  of  action  is  at  an  infinite  distance,  just  as  in 
kinematics  a  rotation  about  an  axis  at  infinity  receives  the 
special  name  of  translation,  and  an  angular  velocity  about  an 
axis  at  infinity  is  called  a  velocity  of  translation. 

It  has  been  shown  in  Kinematics,  Arts.  64,  65,  that  two  equal 
and  opposite  rotations  about  parallel  axes  produce  a  translation, 
and  in  Kinematics,  Art.  256,  that  two  equal  and  opposite  angular 
velocities  about  parallel  axes  produce  a  velocity  of  translation  ; 
similarly,  two  equal  and  opposite  forces  along  parallel  lines  form 
a  new  kind  of  quantity  called  a  couple  of  forces,  or  simply  a  couple. 

While  rotations,  angular  velocities,  and  forces  are  represented 
by  rotors,  i.e.  by  vectors  confined  to  definite  lines,  translations, 
velocities  of  translation,  and  couples  have  for  their  geometrical 
representatives  vectors  not  confined  to  particular  lines. 

Just  as  in  the  case  of  couples  of  infinitesimal  rotations  and  of 
couples  of  angular  velocities,  the  vector  representing  a  couple 


8o 


STATICS. 


[134. 


of  forces  has  for  its  magnitude  and  sense  those  of  the  moment 
of  the  couple,  and  for  its  direction  that  perpendicular  to  the 
plane  of  the  couple. 

It  is  due  to  this  analogy  between  the  two  fundamental  con- 
ceptions that  a  certain  dualism  exists  between  the  theories  of 
statics  and  kinematics,  so  that  a  large  portion  of  the  theory  of 
kinematics  of  a  rigid  body  might  be  made  directly  available  for 
statics  by  simply  substituting  for  angular  velocity  and  velocity 
of  translation  the  corresponding  ideas  of  force  and  couple. 

134.  When  any  number  of  couples  act  on  a  rigid  body  their 
resultant  can  readily  be  found.     Representing  each  couple  by 
its  vector,  we  have  only  to  combine  these  vectors  by  geometrical 
addition.     In  the  particular  case  when    the   couples  all  lie  in 
parallel  planes,  or  in  the  same  plane,  their  vectors  may  be  taken 
in  the  same  line,  and  add,  therefore,  algebraically. 

Hence,  the  resultant  of  any  number  of  couples  is  a  single  cotiple 
whose  vector  is  the  geometric  sum  of  the  vectors  of  the  given  couples. 

Conversely,  a  couple  can  be  resolved  into  components  by 
resolving  its  vector  into  components. 

135.  To  combine  a  single  force  P  with  a  couple  (F,  p)  lying 

in  the  same  plane  it  is  only  nec- 
essary to  place  the  couple  in  its 
plane  into  such  a  position  (Fig. 
35)  that  one  of  its  forces,  say 
—  F,  shall  lie  in  the  same  line 
and  in  opposite  sense  with  the 
single  force  P,  and  to  transform 
the  couple  (F,  /)  into  a  couple 
(P,  p'),  by  Art.  131,  so  that  Fp 
=  Pp'.  The  original  single  force 
P  and  the  force  — P  of  the  trans- 
formed couple  destroying  each 
other  at  A,  there  remains  only 
the  other  force  P,  at  A',  of  the  transformed  couple  which  is  par- 


-P 


-F 


Fig.  35. 


! 


I37-]  THEORY   OF   COUPLES.  Si 

allel  and  equal  to  the  original  single  force  P,  and  has  the  distance 

itfi  i$  pllf-vv  i 

from  it. 

Hence,  a  couple  and  a  single  force  in  the  same  plane  are 
together  equivalent  to  a  single  force  equal  and  parallel  to,  and  of 
the  same  sense  with,  the  given  force,  but  at  a  distance  from  it 
which  is  found  by  dividing  the  moment  of  the  couple  by  the 
single  force. 

136.  Conversely,  a  single  force  P  applied  at  a  point  A  of  a 
rigid  body  can  always  be  replaced  by  an  equal  and  parallel  force 
P  of  the  same  sense,  applied  at  any  other  point  A'  of  the  same  body, 
in  connection  with  the  couple  formed  by  P  at  A  and  —P  at  A'. 

137.  The  proposition  of  Art.  135  applies  even  when  the  force 
lies  in  a  plane  parallel  to  that  of  the  couple,  since  the  couple  can 
be  transferred  to  any  parallel  plane  without  changing  its  effect. 

If  the  single  force  intersects  the  plane  of  the  couple,  it  can 
be  resolved  into  two' components,  one  lying  in  the  plane  of  the 
couple,  while  the  other  is  at  right  angles  to  this  plane.  On 
the  former  component  the  couple  has,  according  to  Art.  135,  the 
effect  of  transferring  it  to  a  parallel  line.  We  thus  obtain 
two  non-intersecting,  or  skew,  forces  at  right  angles  to  each  other. 

Let  P  be  the  given  force,  and  let  it  make  the  angle  «  with  the 
plane  of  the  given  couple,  whose  force  is  F  and  whose  arm 
is/.  Then/*  sin  a  is  the  component  at  right  angles  to  the 
plane  of  the  couple,  while  P  cos  a  combines  with  the  couple 
whose  moment  is  Fp  to  a  force  Pcosa  in  the  plane  of  the 
couple;  this  force  Pcosa  is  parallel  to  the  projection  of  P  on 

the  plane,  and  has  the  distance  — — £—  from  this  projection. 

Pcosa 

Hence,  in  the  most  general  case,  the  combination  of  a  single 
force  and  a  couple  can  be  replaced  by  the  combination  of  two 
single  forces  crossing  each  other  at  right  angles ;  it  can  be 
reduced  to  a  single  force  only  when  the  force  is  parallel  to  the 
plane  of  the  couple. 

PART   II — 6 


82  STATICS.  [138. 

138.   Exercises. 

(1)  Show  that  the  moment  of  a  couple  can  be  represented  by  the 
area  of  the  parallelogram  formed  by  the  two  forces  of  the  couple,  or 
by  twice  the  area  of  the  triangle  formed  by  joining  any  point  on  the 
line  of  one  of  the  forces  to  the  ends  of  the  other  force. 

(2)  Show  that  the  sum  of  the  moments  of  two  forces  forming  a  couple 
is  the  same  for  any  point  in  the  plane  of  the  couple. 

(3)  Show,  by  means  of  Arts.  129-131,  how  to  combine  any  number 
of  couples  situated  in  the  same  plane,  or  in  parallel  planes. 

(4)  Find  the  resultant  of  two  couples  situated  in  non-parallel  planes, 
without  using  the  vectors  of  the  couples. 


I39-]  PLANE   STATICS.  83 

V.    Plane  Statics. 

I.    THE    CONDITIONS    OF    EQUILIBRIUM. 

139.  Suppose  a  rigid  body  to  be  acted  upon  by  any  number 
of  forces,  all  of  which  are  situated  in  the  same  plane.  To 
reduce  such  a  plane  system  of  forces  to  its  simplest  form  the 
proposition  of  Art.  136  may  be  used.  This  proposition  allows 
us  to  transfer  all  the  forces  to  a  common  origin,  by  introducing, 
in  addition  to  each  force,  a  certain  couple  in  the  same  plane. 
The  concurrent  forces  can  then  be  combined  into  their  result- 
ant by  geometric  addition,  or  by  forming  their  force  polygon 
(Art.  98) ;  and  the  couples  lying  all  in  the  same  plane  combine 
by  algebraic  addition  of  their  moments  into  a  resultant  couple 
(Art.  134). 

Thus,  let  F  (Fig.  36)  be  one  of  the  forces  of  the  given  plane 
system,  P  its  point  of  application.  Selecting  any  point  O  in 
the  plane  as  origin,  apply  at 
O  two  equal  and  opposite 
forces  F,  —F,  each  equal 
and  parallel  to  the  given 
force  F\  and  let  /  be  the 
perpendicular  distance  of 
the  origin  O  from  the  line 

of  action  of  the  given  force  Fig.  36. 

'F.      The   force   F  at   P   is 

equivalent  to  the  force  F  at  O  in  connection  with  the  couple 
formed  by  .Fat  P  and  —  Fat  O  ;  the  moment  of  this  couple  is 
Fp,  its  vector  is  perpendicular  to  the  plane  of  the  system. 

Proceeding  in  the  same  way  with  every  force  of  the  given 
system,  all  forces  are  transferred  to  the  common  origin  O. 
The  whole  system  is  therefore  equivalent  to  their  resultant  R 
passing  through  O,  in  connection  with  the  resulting  couple 


84  STATICS.  [140. 

140.  The  given  system  of  forces  will  be  in  equilibrium  if  the 
following  two  conditions  of  equilibrium  are  fulfilled  : 

Jt  =  o,     H=o. 

It  will  be  noticed  that  the  moment  Fp  of  the  couple  intro- 
duced by  transferring  the  force  F  to  the  point  O  is  the  moment 
of  the  force  Fwith  respect  to  this  point  O. 

Hence,  a  plane  system  of  forces  is  in  equilibrium  if  (a)  its 
resultant  is  zero,  and  (b)  the  algebraic  sum  of  the  moments  of  all 
its  forces  is  zero  with  respect  to  any  point  in  its  plane. 

141.  It  is  evident  that  the  magnitude  and  direction  of  the 
resultant  R  do  not  depend  on  the  selection  of  the  origin  O. 
But  the  position  of  this  resultant  and  the  magnitude  of  the 
resulting  couple  //  will  in  general  differ  for  different  points 
selected  as  origin.      Indeed,  the  origin  can  be  so  taken  as  to 
make  the  couple  H  vanish  (unless  the  resultant  R  be  zero)  ; 
that  is,  the  whole  system  can  be  reduced  to  a  single  resultant. 

To  do  this  (see  Art.  135),  it  is  only  necessary,  after  determin- 
ing R  and  H  for  some  point  O,  to  transfer  R  to  a  parallel  line 
at  such  a  distance  r  from  its  original  position  as  to  make  the 
moment  Rr  of  the  couple  introduced  by  the  transfer  equal  and 
opposite  to  the  moment  ^Fp  ;  i.e.  we  must  take  (Art.  135) 


The  line  along  which  this  single  resultant  acts  is  called  the 
central  axis  of  the  given  system  of  forces. 

142.  For  a  purely  analytical  reduction  of  a  plane  system  of 
forces  the  system  is  referred  to  rectangular  axes  Ox,  Oy,  arbi- 
trarily assumed  in  the  plane  (Fig.  37).  Every  force  /MS  resolved 
at  its  point  of  application  P  (x,  y)  into  two  components  X,  Y, 
parallel  to  the  axes,  so  that 

F=/7sin«, 


PLANE   STATICS. 


-X 


.«  being  the  angle  made  by  .Fwith  the  axis  Ox.  At  the  origin  O 
two  equal  and  opposite  forces  X,  —  X  are  applied  along  Ox,  and 
two  equal  and  opposite 
forces  F,  —  Y  along  Oy. 
Thus,  X  at  P  is  equivalent 
to  X  at  O  in  combination 
with  the  couple  formed  by 
X  at  P  and  -JTat  <9 ;  the 
moment  of  this  couple  is 
•evidently  —yX.  Similarly, 
Fat  P  is  replaced  by  Fat 
O  in  combination  with  a 
•couple  whose  moment  is  xY. 

The  force  Fat  P  is  therefore  equivalent  to  the  two  forces  X,  Y 
.at  O  in  combination  with  a  couple  whose  moment  is  xY—  yX. 

Proceeding  in  the  same  way  with  every  given  force,  we  obtain 
a  number  of  forces  X  along  Ox  which  can  be  added  algebrai- 
cally into  2,Xf  and  a  number  of  forces  Y  along  Oy  which  give 
.2  Y.  These  two  rectangular  forces  form  the  resultant 


whose  direction  is  given  by 
tan«  = 


where  a  is  the  angle  between  Ox  and  R. 

In  addition  to  this,  we  obtain  a  number  of  couples  xY—yX 
ivhose  algebraic  sum  forms  the  resulting  couple 


143.  The  whole  system  is  thus  found  equivalent  to  a 
resultant  force  R  in  combination  with  a  resultant  couple  H  in 
the  same  plane  with  R.  The  conditions  of  equilibrium  R  =  o, 
H=o  (Art.  140)  can  therefore  be  expressed  analytically  by  the 
three  equations 


86  STATICS.  [144. 

144.  If  R  be  not  zero,  R  and  H  can  be  combined  into  a  single 
resultant  R'  equal  and  parallel  to  R  at  the  distance  —H/R  from 
it  (see  Art  141).  The  equation  of  the  line  of  this  single  result- 
ant R'y  i.e.  the  central  axis  of  the  system  of  forces,  is  found  by 
considering  that  it  makes  the  angle  a  with  the  axis  of  x  and  that 
its  distance  from  the  origin  is 


H/R  =  2  (x  Y- 
Hence  its  equation  is 

f  -  2F-17  •  2AT-  2  (x  Y-yX)  =  o. 
If  R  =  o,  the  system  is  equivalent  to  the  couple 


unless  H  itself  be  also  zero,  in   which  case  the  system  is  in 
equilibrium. 

145.   The  same  results  can  be  obtained  by  a  transformation 


of  co-ordinates.  Let  R  =  V(S^T)2-f  (2F)2  and  H=  2  (x  Y-yX  ) 
be  the  resultant  force  and  couple  for  a  point  O  as  origin.  If 
some  other  point  O',  whose  co-ordinates  with  respect  to  O  are 
(•  ,  77,  be  taken  as  new  origin  and  x\  y1  be  the  co-ordinates  of  the 
point  of  application  P  of  the  force  F  for  parallel  axes  through 
O\  the  resultant  R  remains  the  same  while  the  resulting  couple 
becomes 


Hence  this  new  couple  will  vanish  whenever  the  origin  O'(%,  rj) 
is  taken  on  the  straight  line  whose  equation  referred  to  the 
original  axes  is 

77-77=0. 


This  equation  of  the  central  axis  agrees  with  the  equation  found 
in  Art.  144;  it  represents  the  line  of  action  of  the  single 
resultant  to  which  the  system  can  be  reduced. 


147.]  PLANE   STATICS.  8/ 

146.  The  following  examples  will  illustrate  the  application  of 
the  conditions  of  equilibrium.     To  establish  these  conditions  in 
any  particular  problem  it  will  generally  be  found  best  to  resolve 
the  forces  along  two  rectangular  directions  and  equate  the  sums 
of  the  components  to  zero;  and  then  to  "take  moments,"  i.e. 
equate  to  zero  the  sum  of  the  moments  of  all  the  forces  with 
respect  to  some  point  conveniently  selected  as  origin. 

147.  A  homogeneous  straight  rod  AB=  2!  (Fig.  38)   of  weight  W 
rests  with  one  end  A  on  a  smooth  horizontal  plane  AH,  and  with  the 
point  E(AE  =  e)  on  a  cylindrical  support,  the  axis  of  the  cylinder  being 
at  right  angles  to  the  vertical  plane  containing  the  rod.     Determine  what 
horizontal  force  F  must  be  applied  at  a  given  point  F  of  the  rod  (AF 
=  f>e)  to  keep  the  rod  in  equilibrium  when  inclined  to  the  horizon  at  an. 
angle  6. 


The  rod  exerts  a  certain  unknown  pressure  on  each  of  the  supports  at 
A  and  E,  in  the  direction  of  the  normals  to  the  surfaces  of  contact,  pro- 
vided there  be  no  friction,  as  is  here  assumed.  The  supports  may 
therefore  be  imagined  removed  if  forces  A,  E,  equal  and  opposite  to 
these  pressures,  be  introduced  ;  these  forces  A,  E  are  called  the 
reactions  of  the  supports.  The  rod  itself  is  here  regarded  as  a  straight 
line  ;  its  weight  W  is  applied  at  its  middle  point  C. 

Taking  A  as  origin  and  AH  as  axis  of  x,  the  resolution  of  the  forces 
gives 

o,  (i) 


o.  (2) 

Taking  moments  about  A,  we  find 

E-e-W-  /cos  &  -  /Vsin  0  =  o.  (3) 


88  STATICS.  [148. 

Eliminating  F  from  (i)  and  (3),  we  have 


__ 

e  —/sin2  0 
hence  from  (2)  , 


e—/sm2O 
and  finally  from  (i), 


148.  A  cylinder  of  length  2!  and  radius  r  rests  with  the  point  A  of 
the  circumference  of  its  lower  base  on  a  horizontal  plane  and  with  the 
point  B  of  the  circumference  of  its  upper  base  against  a  vertical  wall. 
The  vertical  plane  through  the  axis  of  the  cylinder  contains  the  points  A, 
B,  and  is  perpendicular  to  the  intersection  of  the  vertical  wall  and  the 
horizontal  plane.  If  there  be  no  friction  at  A  and  B,  what  horizontal 
force  F  applied  at  A  will  keep  the  cylinder  in  equilibrium  ?  When  is  this 
force  F  =  o  ? 

Let  G  be  the  centre  of  gravity  of  the  cylinder ;  W  its  weight ;  A,  B 
the  reactions   at  A,   B ;    and   9  the  given 
angle  between  AB  and  the  horizontal  plane. 
B  Then  B  —  F—  o,  A  —  W—  o,  and  taking 

moments  about  A, 


If  either  the  dimensions  of  the  cylinder,  or  the  angle  6,  be  such  as  to 
make  tan  B  =  l/r,  no  force  F  will  be  required  to  maintain  equilibrium  ; 
G  and  A  will  then  lie  in^the  sarne  vertical  line. 

149.  The  homogeneous  rod  AB  =  2  1  of  weight  W  is  jointed  at  A,  so 
as  to  turn  about  A  in  a  vertical  plane.  A  string  BC  attached  to  the 
end  B  of  the  rod  runs  at  C  over  a  smooth  pulley^  and  carries  a  weight  P. 
The  axis  of  the  pulley  C  is  parallel  to,  and  in  the  same  vertical  plane 


150.] 


PLANE   STATICS. 


89 


with,  the  axis  of  the  joint  A  ;  AC  =  h.     Find  the  position  of  equilibrium 
and  the  pressure  on  the  axis  of  the  joint  A.     (Fig.  40.) 

To  reduce  to  a  purely  statical  problem,  cut  the  string  between  B  and 
C  and  introduce  the  tension,  which  is  =  P-}  also,  replace  the  pressure 
A  by  its  horizontal  and  vertical  components  Ax,  Ay.  Then,  if  ^  ACB 
=  </>,  %.jBAC=0,  the  conditions  of  equilibrium  give 

Ax  =  Ps'm<f>,     Ay  =  W-  /'cos  <£, 


From  the  last  equation, 

sm>=/^ 
sin0  ~  h    P9 

while  from  the  triangle  ABC, 
sin  <£       2  / 


sin0 


hence 
sented 


=  zhP/W,  i.e.  if  we  take  ^  to  represent  W,  Pwill  be  repre- 
£C. 

For  the  total  pressure  A  we  have 


i.e.  A  is  the  third  side  of  a  triangle  having  W  and  P  for  the  two  other 
sides,  and  <£  for  the  included  angle.  The  magnitude  of  A  is  therefore 
represented  by  the  median  from  A  in  the  triangle  ABC  on  the  same 
scale  on  which  Wis  represented  by  h.  But  this  median  gives  also  the 
direction  of  A ;  for  we  have 

k     BC  CQZ& 
A,,      W- 


150.  A  weightless  rod  AB  rests  without  friction  on  two  planes 
inclined  to  the  horizon  at  angles  a,  j3}  and  carries  a  weight  W  at  the 
point  D.  The  intersection  (C)  of  these  planes  is  horizontal  and  at  right 
angles  to  the  vertical  plane  through  AB.  Find  the  inclination  0  of  ^AB 
to  the  horizon,  and  the  pressures  at  A  and  B.  (Fig.  41.) 


9o 


STATICS. 


[If*- 


As  there  are  only  three  forces,  viz.  the  weight  Wand  the  reactions  A 
and  B,  their  lines  must  intersect  in  a  point  E.     Resolving  horizontally 

and  vertically,  we  have 


A  sin  a  =  B  sin  /?, 


W 


W. 


whence  A= 


sin  a 


sin(« 


Taking  moments  about  D,  we  find 
with  AD  =  a,  DB  =  b, 


W 


Fig.  41. 


A  -  a  sin  DAE  =  B 
or  ^«  cos(«  +  0)  =  Bb  cos(/J  -  0)  ; 
to  eliminate  A  and  B,  divide  by  the  first  equation  above  : 


sn  « 
solving  for  0,  we  finally  obtain 


sn 


151.   Exercises. 

(1)  A   homogeneous   rod   yl#=2/=8ft.,   weighing    W=2Q  Ibs., 
rests  with  one  end  A  on  a  horizontal  plane  AH,  and  with  the  point  E 
on  a  support  whose  height  above  AH  is  Z^£  =  h  =  3  ft.     A  horizontal 
cord  AD  =  d  =  4  ft.  holds  the  rod  in  equilibrium.     Find  the  tension  T 
of  this  cord,  and  the  reactions  at  A  and  E. 

(2)  A  weightless  rod  AB  of  length  /  can  turn  freely  about  one  end  A 
in  a  vertical  plane.     A  weight  W  is  suspended  from  a  point  C  of  the 
rod  ;  A  C  =  c.     A  string  BD  attached  to  the  end  B  of  the  rod  holds  it 
in  equilibrium  in  a  horizontal  position,  the  angle  ABD  being  «=  150°. 
Find  the  tension  T  of  the  string  and  the  resulting  pressure  A  on  the 
hinge  at  A. 


I5i.]  PLANE   STATICS.  9! 

(3)  A  uniform  rod  AB  —  2 1  of  weight  W  rests  with  its  upper  end  A 
against  a  smooth  vertical  wall,  while  its  lower  end  B  is  fastened  by  a 
string  of  given  length,  BC=  2  b,  to  a  point  C  in  the  wall.     The  rod  and 
the  string  are  in  the  vertical  plane  at  right  angles  to  the  wall.     Find  the 
position  of  equilibrium,  i.e.  the  angle  $  =  ACB,  the  tension  7"of  the 
string,  and  the  pressure  A  against  the  wall. 

(4)  A  uniform  rod  AB  =  2 /of  weight  W  rests  with  one  end  A  on  a 
smooth  horizontal  plane  AC,  with  the  other  end  B  against  a  smooth 
vertical  wall  BC,  the  vertical  plane  through  AB  being  at  right  angles  to 
the  intersection  C  of  the  wall  with  the  horizontal  plane.    The  rod  is  kept 
in  equilibrium  by  a  string  EC.     Find  the  tension  Tof  this  string  if  the 
angles  CAB  =  0  and  EC  A  =  <f>  are  given. 

(5)  A  weightless  rod  AB =  /  can  revolve  in  a  vertical  plane  about  a 
hinge  at  A  ;  its  other  end  B  leans  against  a  smooth  vertical  wall  whose 
distance  from  A  is  AD  =  a.     At  the  distance  A  C  =  c  from  A,  a  weight 
W  is  suspended.     Find  the  horizontal  thrust  Ax  at  A  and  the  normal 
pressures  Ay  and  B  at  A  and  B. 

(6)  The  same  as  (5)  except  that  at  B  the  rod  rests  on  a  smooth  hori- 
zontal cylinder  whose  axis  is  at  right  angles  to  the  vertical  plane  through 
AB.     In  which  of  the  two  problems  is  the  horizontal  thrust  Ax  at  A 
least? 

(7)  The  lower  end  A  of  a  smooth  uniform  rod  AB  of  weight  Wrests 
on  a  smooth  horizontal  plane  making  an  angle  0  with  it.    At  the  point  C  it 
rests  on  a  smooth  cylinder  whose  axis  is  horizontal  and  at  right  angles  to 
the  vertical  plane  through  the  rod ;  at  D  the  rod  is  pressed  upon  by 
another  smooth  cylinder  whose  axis  is  parallel  to  that  of  the  cylinder 
at  C.     Determine  the  reactions  at  A,  C,  D,  if  W,  0,  AB  =  2  /,   CD  =  a 
are  given. 

(8)  A  smooth  weightless  rod  AB  =  /rests  at  C  on  a  smooth  horizon- 
tal cylinder  whose  axis  is  at  right  angles  to  the  vertical  plane  through 
the  rod ;  its  lower  end  A  leans  against  a  smooth  vertical  wall  whose  dis- 
tance  from    C  is    CD  —  a ;   from    its  upper  end  B  a  weight  W  is 
suspended.     Determine  the  distance  AC  =  x  for  equilibrium,  and  the 
reactions  at  A  and  C. 

(9)  A  uniform  rod  of  weight  W\s  hinged  at  its  lower  end  A,  while  its 
upper  end  B  leans  against  a  smooth  vertical  wall.     The  rod  is  inclined 
at  an  angle  0  to  the  vertical,  and  carries  three  weights,  each  equal  to 
w,  at  three  points  dividing  the  rod  into  four  equal  parts.      Determine 
the  pressure  on  the  wall  and  the  reaction  of  the  hinge. 


.92  STATICS.  [152. 

(10)  A  homogeneous  rod  AB  =  2  /  of  weight  Crests  with  one  end  A 
on  the  inside  of  a  fixed  hemispherical  bowl  of  diameter  2  a  and  leans  at 
C  on  the  horizontal  rim  of  the  bowl,  so  that  the  other  end  B  is  outside. 
Determine  the  inclination  to  the  horizon  9  in  the  position  of  equilibrium. 


2.     STABILITY. 

152.  The  equilibrium  of  the  forces  acting  on  a  rigid  body  may 
subsist  while  the  body  is  in  motion.     Thus,  if  the  motion  con- 
sist in  a  mere  translation  with  constant  velocity,  the  equilibrium 
will  not  be  disturbed  during  the  motion  if  the  forces  remain 
equal  and  parallel  to  themselves. 

If,  however,  the  body  be  subjected  to  a  rotation,  this  will  in 
general  not  be  the  case.  The  present  considerations  are  re- 
stricted to  the  case  of  plane  motion ;  the  forces  are  supposed 
to  lie  in  the  plane  of  the  motion  and  to  remain  equal  and 
parallel  to  themselves  and  applied  at  the  same  points  of  the 
body. 

153.  Let  A^AZ  (Figs.  42  and  43)  be  a  rigid  rod  having  two 
equal  and  opposite  forces  Flt  F2  applied  at  its  extremities  in  the 


*-*      A?     i     K     '' 


Fig.  43. 


direction  of  the  line  A^A^.  Let  this  rod  be  turned  by  an  angle 
<f>  about  an  axis  at  right  angles  to  A^A^.  In  the  new  position 
the  forces  Fv  F2,  instead  of  being  in  equilibrium,  form  a  couple 
whose  moment  is  ±Fl'  A^A^  sin  <f>. 

If  in  the  original  position  of  the  rod  the  forces  tend  to  increase 
the  distance  A^2  (Fig.  42),  the  couple  in  the  new  position  will 


I55-]  STABILITY.  93 

tend  to  bring  the  rod  back  to  its  position  of  equilibrium.  In 
this  case  the  original  position  of  the  rod  is  said  to  be  a  position 
of  stable  equilibrium.  The  effect  of  the  earth's  magnetism  on 
the  needle  of  a  compass  offers  a  familiar  example. 

If,  however,  in  the  original  position  the  forces  tend  to  diminish 
the  distance  A1A2  (Fig.  43),  the  couple  arising  after  displace- 
ment tends  to  increase  the  displacement  and  thus  to  remove  the 
rod  still  farther  from  equilibrium.  The  original  position  in  this 
case  is  said  to  be  one  of  unstable  equilibrium.  The  weight  rest- 
ing on  a  vertical  post  and  the  reaction  of  the  support  on  which 
the  rod  stands  may  be  taken  as  an  illustration. 

Finally,  a  third  case  would  arise  if  the  forces  Flt  F2,  being 
still  equal  and  opposite,  were  applied  at  one  and  the  same  point 
of  the  rod.  The  forces  would  then  remain  in  equilibrium  after 
any  displacement  of  the  rod ;  such  equilibrium  is  called  neutral 
or  astatic. 

154.  These  different  cases  of  equilibrium  can  be  distinguished 
by  the  algebraic  sign  of  the  product  A^A^  •  F1  =  A2Al  •  F2,  which 
is    negative   for    stable    equilibrium,   since  A^A^  and   F1    have 
opposite    sense    (Fig.    42),    positive    for    unstable    equilibrium 
(Fig.  43),  and  indeterminate  (since  A^^o)  for  neutral  equi- 
librium. 

It  is  to  be  noticed  that  these  considerations  will  hold  whether 
the  rotation  of  angle  </>  take  place  in  the  positive  or  negative 
sense.  But  they  hold  only  within  certain  limits  for  the  angle  of 
rotation.  Thus,  in  the  example  illustrated  by  Figs.  42  and  43, 
when  <f>  reaches  the  value  TT,  the  nature  of  the  equilibrium  is- 
changed. 

155.  Strictly   speaking,    investigations   of    stability   are   not 
purely  statical,  but  require  a  kinetic  examination  of  the  subse- 
quent motion.     However,  the  principles  of  statics  are  sufficient 
to  determine  the  nature  of  the  equilibrium  for  infinitesimal  dis- 
placements, i.e.  when  only  the   initial   motion  of   the  body  is- 
considered. 


•94 


STATICS. 


[156. 


The  theory  of  astatic  equilibrium  forms  a  special  branch  of 
mechanics  called  astatics;  its  object  is  to  determine  the  condi- 
tions under  which  a  system  of  forces  acting  on  a  rigid  body 
remains  in  equilibrium  when  the  body  is  subjected  to  any 
•displacement  while  the  forces  remain  applied  at  the  same  points 
•of  the  body  and  retain  their  magnitude,  direction,  and  sense. 

156.  The  equilibrium  of  forces  acting  on  one  and  the  same 
point  is  evidently  always  astatic. 

In  the  case  of  a  plane  system  of  forces  acting  on  a  plane 
figure  in  its  plane,  the  only  displacement  that  need  be  consid- 
ered is  a  rotation  about  an  axis  at  right  angles  to  the  plane. 
For  every  displacement  of  a  plane  figure  in  its  plane  can  be 
reduced  to  a  rotation  about  a  certain  centre  in  the  plane. 

Instead  of  turning  the  body  or  plane  figure  by  an  angle  c/>, 
we  may  turn  all  the  forces  about  their  points  of  application  by 
the  same  angle  in  the  opposite  sense. 

157.  If  the  plane  system  consists  of  two  forces  in  equilibrium, 
they  must  be  equal  and  opposite,  and  act  in  the  same  line  ; 

this    case   has    been    considered    in 
Art.  154. 

If  there  be  three  forces  Flt  Fz,  F3 
in  the  same  plane  in  equilibrium, 
applied  at  the  points  Av  A2,  A3, 
they  must  meet  in  a  point  Ot  and 
fulfil  the  parallelogram  law. 

After  turning  each  force  about  its 
point  of  application  by  the  same 
angle,  the  forces  will,  in  general, 
cease  to  intersect  in  a  point,  and 

hence  to  be  in  equilibrium.  If,  however,  the  original  meeting 
point  O  of  the  forces  be  situated  on  the  circle  described  through 
Av  A2,  AB  (Fig.  44),  the  forces  will  continue  to  intersect  at  some 
point  of  this  circle  when  turned  through  some  angle,  because 
the  angles  between  the  forces  remain  constant.  ^ 


1 59-]  STABILITY.  95 

Hence,  three  forces  F15  F2,  F3  in  the  same  plane,  applied  at 
points  Aj,  A2,  A3,  are  in  astatic  equilibrium  if  they  meet  in  a 
point  O  situated  on  the  circle  passing  through  Alf  A2,  A3. 

The  condition  of  equilibrium  of  the  three  forces  also  requires 
that 

F,  F,  F. 


sin  (F2F3)     sin  (/r3/71)     sin 

by  the  property  of  the  circle  (Fig.  44),  we  have  ^  (F2F3)  =  A  v 
%.(F8Fi)  =  A2,  '^.(F1F2)  =  7r  —  A3  ;  and  as  the  sines  of  these  angles 
of  the  triangle  A^A^A^  are  proportional  to  the  opposite  sides, 

we  have 

F,          F» 

1 =  A. 

^2^3       ^3^ 

i.e.  three  forces  in  astatic  equilibrium  are  to  each  other  as  the  sides 
of  the  triangle  formed  by  their  points  of  application. 

158.  The  results  of  the  preceding  article  can  be  interpreted 
from  a  somewhat  different  point  of  view.    Let  two  of  the  forces, 
F1  and  F2,  be  given,  and  let  it  be  required  to  determine  their 
resultant  for  astatic  equilibrium.     This  resultant  F9  must  evi- 
dently pass  through  a  definite  point  A3  of  the  circle  described 
through  the  points  of  application  Av  A2  of  the  given  forces  and 
their  intersection  O.    This  point  As,  through  which  the  resultant 
must  pass,   howsoever  the  two  given  forces   be  turned   about 
Av  A2,  is  called  the  centre  of  the  forces. 

If  the  two  given  forces  be  parallel,  the  point  O  lies  at  infinity, 
and  the  circle  through  AI}  A2,  O  becomes  the  straight  line 
A^A^.  The  point  A3  is  therefore  situated  on  this  line  and 
divides  the  distance  A-^A2  in  the  inverse  ratio  of  the  forces  Fv 
F2,  by  Art.  157.  Compare  Art.  no. 

159.  These  results  are  readily  generalised.     Any  plane  sys- 
tem of  forces  has  a  centre  unless  the  resultant  be  zero.     To 
find  the  centre  we  have  only  to  combine  the  forces  in  succession, 
i.e.  to  find  the  centre  of   two  of  the  forces,  then  the  centre 
of  their  resultant  and  a  third  force,  etc. 


96  STATICS.  [160. 

It  has  been  shown  in  Arts.  141  and  144  that  a  plane  system 
of  forces  whose  resultant  does  not  vanish  can  always  be  reduced 
to  a  single  resultant  R  whose  line  is  called  the  central  axis  of 
the  system.  It  appears  now  that  if  the  forces  be  all  turned  by 
the  same  angle  6  about  their  points  of  application,  the  line  of  the 
resultant,  or  the  central  axis,  will  turn  about  a  certain  fixed 
point  called  the  centre  of  the  system.  "  For  a  system  of  parallel 
forces  the  existence  of  such  a  centre  has  already  been  proved  in 
Art.  1  10. 

160.  Analytically,  the  centre  of  a  plane  system  of  forces  is 
found  as  the  intersection  of  the  two  positions  of  the  central  axis 
before  and  after  any  displacement  of  the  plane  figure,  or  body, 
on  which  the  forces  act. 

By  Arts.  144  and  145  the  equation  of  the  central  axis  is 

^Y>Z-^"n-^(xY-yX}=o.  (i) 

Let  the  figure  with  the  axes  of  co-ordinates  be  turned  through 
an  angle  0  about  an  axis  through  the  origin  perpendicular  to  its 
plane,  while  the  forces  keep  their  original  directions.  The  cen- 
tral axis  of  the  forces  in  the  new  position  will  have  an  equa- 
tion of  the  same  form  as  before,  in  which,  however,  x,  y,  f,  77  are 
referred  to  the  new  system  of  co-ordinates.  To  find  the  equa- 
tion of  the  central  axis  in  the  old  co-ordinates,  we  have  to 
substitute  x  cos  </>  —  y  sin  </>  for  x,  ;rsin  (ft+y  cos  0  forj>,  and 
similarly  for  f,  77.  This  gives 


sn 
—  2  \(x  cos  <f>  —y  sin  <£)  Y—  (x  sin  <f>  +y  cos  <j))X  ]  =  o, 

or  collecting  the  terms  containing  cos  <f>  and  sin  <£,  respectively, 


0=0.  (2) 

The  centre  being  the  intersection  of  the  lines  (i)  and  (2),  its 


i62.]  STABILITY.  97 

co-ordinates  are  found  by  solving  these  equation  for  f  and  77,  or 
.e  the  coefficient  of  cos  <f>  in  (2)  vanishes  by  (i),  by  solving 
-.he  equations 

(3) 
'(4) 

Putting,  for  shortness,  V(2X)2  +  (£Y)2=R,  t(xY— yX)  —  H, 
2,(xX+yY)  =  K,  we  find  the  co-ordinates  of  the  centre, 


"= 


161.  By  the  rotation  of  the  figure,  the  magnitude  of  the 
resultant  R  of  the  system  is  of  course  not  changed.  But 
the  resulting  couple  H  for  the  origin,  or  what  amounts  to  the 
.same,  the  moment  of  the  system  about  the  origin,  is  changed  and 
becomes,  by  Art.  160, 


H1  =  2  [(x  cos  (/>  ~y  sin  <f>)  Y—  (x  sin  <£  +y  cos 
yX)  -  cos  </>-2  (xX+y  F)  -  sin  0 
—  K  sm  </>.  (6) 


This  couple  //"'  vanishes  if  the  figure  be  turned  through  an 
•angle  </>  determined  by  the  equation 

tan<£=^.  (7) 

162.  If  the  system  of  forces  be  originally  in  equilibrium,  we 
have  2X=o,  2F=o,  ^(xY-yX}=o  (Art.  143).  Hence  after 
turning  the  figure  through  an  angle  <£,  the  forces  will  be  equiva- 
lent to  the  couple 

H'=-Ksin<l>.  (8) 

This  couple  has  its  greatest  value  when  0  =  ?r/2;  it  vanishes 
only  when  <£  =  TT,  in  which  case  the  system  will  again  be  in 
equilibrium. 

PART   II  —  7 


g8  STATICS.  [163. 

163.  The  stability  of  a  plane  system  in  equilibrium  depends. 
on  the  algebraic  sign  of  the  quantity,  ~ecl 

K=${?X+yY),  (9, 

which  can  therefore  be  called  the  stability  function.  If  this  func- 
tion be  positive,  the  equilibrium  is  stable ;  if  it  be  negative,  the 
equilibrium  is  unstable  ;  finally,  if  K=o,  the  system  is  astatic, 
and  the  equilibrium  is  neutral. 

The  proof  follows  at  once  from  equation  (8).  This  equation 
shows  that,  for  a  positive  K,  the  moment  of  the  couple  to  which 
the  system  becomes  equivalent  when  the  figure  is  turned 
through  an  angle  </>  has  a  sign  opposite  to  that  of  the  angle  </> ; 
hence  this  couple  will  tend  to  turn  the  body  back  into  the 
position  of  equilibrium.  Similarly,  if  K  be  negative,  H1  agrees 
in  sign  with  </>  and  tends  therefore  to  increase  this  angle. 

164.  Exercises. 

(1)  Explain  the  nature  of  the  equilibrium  of  a  body  of  weight  W 
supported  at  a  single  point  according  to  the  position  of  that  point  above 
the  centroid  G,  below  G,  and  at  G  (common  balance). 

(2)  A  homogeneous  rod  AB—  2 /of  weight  W  leans  with  the  lower 
end  A  against  a  vertical  wall  and  rests  with  the  point  C  (A  C  =<:>/) 
on  a  cylindrical  support.     Show  that  the  equilibrium  is  unstable. 

(3)  A  body  of  weight  W  is  placed  on  a  horizontal  plane.     Show  that 
the  equilibrium  is  stable  if  W  meets  the  horizontal  plane  at  a  point 
A  within  the  area  of  contact  and  that  it  is  unstable  if  A  lies  on  the  con- 
tour of  this  area.     If  the  actual  area  of  contact  have  re-entrant  angfes,. 
or  consist  of  several  detached  portions,  the  area  bounded  by  a  thread 
drawn  tightly  around  the  actual  area,  or  areas,  of  contact  must   be 
substituted. 

(4)  An  oblique  cylinder  rests  with  its  circular  base  on  a  horizontal 
plane  in  unstable  equilibrium.     If  the  length  of  its  axis  be  twice  the 
diameter  of  its  base,  what  is  the  inclination  of  the  axis  to  the  horizon  ? 

(5)  Show  how  to  determine  graphically  the  stability  of  a  retaining 
wall  against  toppling  over  the  front  edge  of  the  base,  the  pressure  of  the 
earth  behind  the  wall  being  given  in  magnitude,  direction,  and  position. 


1 66.]  JOINTED   FRAMES.  99 


3.    JOINTED    FRAMES. 

165.  The  equations  of  equilibrium  are  derived  on  the  suppo- 
sitions that  all  the  forces  of  the  given  system  act  on  one  and 
the  same  rigid  body  and  that  this  body  is   perfectly  free  to 
move.     Hence,  in  applying   these  equations  to  determine  the 
equilibrium  of  an  engineering  structure,  a  machine,  etc.,  each 
rigid  body  must    be  considered    separately,  and  the  reactions 
required  to  make  the  body  free  must  be  introduced.     It  will  be 
shown  in  a  subsequent  section  how  the  principle  of  work  makes 
it  possible  to  dispense  with  some  of  these  precautions. 

When  two  rigid  rods  are  connected  by  a  pin-joint  whose  axis 
is  perpendicular  to  the  plane  of  the  rods,  the  action  of  either  rod 
on  the  other  at  the  joint  is  represented  by  a  single  force  whose 
direction  is  in  general  unknown.  Sometimes  considerations  of 
symmetry  will  allow  to  determine  this  direction. 

If  a  rigid  rod,  in  equilibrium,  be  hinged  at  both  ends  and  not 
acted  upon  by  any  other  forces,  the  reactions  of  the  hinges 
must  of  course  be  along  the  rod,  and  must  be  equal  and 
opposite. 

166.  Two  rods  AC,  BC  (Fig.  45)  in  a  vertical  plane,  hinged  together 
at  C,  rest  with  the  ends  A,J$0na  horizontal  plane,  and  carry  a  weight 
W  suspended  from  the  joint  C.     If  the  proper 

weight  of  the  rods  be  neglected,  determine  the 
normal  pressures  Ay,  By  and  the   horizontal 


thrusts  Ax,  Bx  at  A,  B.  k/a       WT     ^\B 


Resolving  the  weight  W  along  C/4,  CB  into 
^A>  ^B  and  considering  the  rod  AC  alone  it 

appears  that  the  total  reaction  at  A  is  along  A  C  and  =  WA ;  hence 
resolving  WA  in  the  horizontal  and  vertical  directions,  Ax  and  Ay  are 
found  ;  similarly  for  BC.  If  a,  ft  be  the  angles  at  A  and  -5  in  the  tri- 
angle ABC,  we  find 

JpA=_cosg_jfr      WB=       cos  a       ^. 


100  STATICS.  [167. 

A  _  ft  _  cos  a  cos  ft  w     A  —  sm  a  cos  P  w      &  _  cos  «  sin  /?  r^ 
x~~sin(«+£)  y~  sin («  +  /?)  *~~  sin  (a  +  0) 

As  the  horizontal  thrusts  at  ^4  and  ^  are  equal,  it  makes  no  difference 
whether  the  rods  be  hinged  to  the  support  at  A  and  B,  or  whether  the 
thrust  is  taken  up  by  lateral  supports,  or  by  a  string  connecting  the  ends 
A,  B  of  the  rods. 

167.    Two  equal  homogeneous  rods  AC,  BC  (Fig.  46)  are  hinged  at 
A,  B,  C  so  as  to  form  a  triangle  whose  height  h  is  vertical  and  whose  base 
AB  =  2  b  is  horizontal.     The  weight  of  each  rod 
being  W,Jind  the  reactions  at  the  joints. 

Owing  to  the  symmetry  of  the  figure,  the  reac- 
tions at  C  must  be  equal  and  opposite  and 
horizontal.  The  rod  AC  is  subject  to  three 
forces  only,  viz.  the  horizontal  reaction  C,  the 
weight  Wj  and  the  reaction  A  •  the  latter  must 
46  *  therefore  pass  through  the  intersection  D  of 

C  and  W. 

If  the  direction  of  W  intersect  AB  at  E  and  the  scale  of  forces  be 
taken  so  as  to  have  W  represented  by  DE  =  h,  DEA  will  be  the  force 
polygon ;  hence  EA  represents  C  and  AD  represents  A  on  the  same 
scale  on  which  W  is  represented  by  h. 

Analytically,  the  reactions  are  found  by  resolving  the  forces  horizon- 
tally and  vertically  and  taking  moments  about  A  : 


whence  C=mWy      A  = 

where  m  == 

zh 

168.  Two  equal  homogeneous  rods  AC,  BC,  each  of  weight  W,  are 
hinged  at  C  ;  their  ends  A,  B  rest  on  a  smooth  horizontal  plane  ;  a  third 
redDE  is  hinged  to  them,  connecting  their  middle  points  (Fig.  47). 

The  plane  AB  being  smooth,  the  reaction  at  A  is  vertical ;  the  reac- 
tion at  C  is  horizontal  owing  to  the  symmetry ;  that  at  D  is  likewise 
horizontal  if  the  weight  of  the  rod  DE  be  neglected,  for  then  this  rod  is 
subject  only  to  the  reactions  at  its  ends. 

Resolving  horizontally  and  vertically  and  taking  moments  about  Z>,  we 
find  in  this  case 


I7i.] 


JOINTED   FRAMES. 


IOI 


where  a  = 


If,  however,  the  weight  w  of  the 
rod  DE  cannot  be  neglected,  we 
have  at  D  a  horizontal  reaction  Dx 
and  a  vertical  reaction  Dv.  The 
equilibrium  of  DE  requires  that 

zD,,  =  w.      Hence    resolving    and 

.  .y  ,    ,  -    , 

taking  moments  as  before,  we  find 


p. 


169.  Exercise. 

(i)  Two  homogeneous  rods  AC,  BC  of  equal  weight,  but  unequal 
length,  are  hinged  together  at  C  while  their  other  ends  are  attached  to 
fixed  hinges  A,  B  in  the  same  vertical  line.  Show  that  the  line  of  action 
of  the  reaction  at  C  bisects  AB. 

170.  A  triangular  frame  formed  of  rigid  rods  is  rigid  as  a 
whole,  even  when  the  connections  are  pin-joints.     A  quadran- 
gular frame  with  pin-joints  becomes  rigid  only  by  the  insertion 
of  a  diagonal. 

The  iron  and  steel  trusses  'used  for  roofs  and  bridges  gener- 
ally consist  of  a  system  of  triangles,  or  quadrangles  with  diago- 
nals, so  that  the  whole  truss  can  be  regarded  as  one  rigid  body, 
at  least  in  first  approximation. 

Any  one  rod,  or  member,  of  the  frame-work  is  thus  acted  upon 
by  two  equal  and  opposite  forces,  i.e,  by  a  stress,  in  the  direction 
of  its  length,  the  external  forces,  including  the  proper  weight, 
being  regarded  as  applied  at  the  joints  only.  If  the  stress  be  a 
tension,  i.e.  if  the  forces  tend  to  stretch  or  elongate  the  mem- 
ber, the  latter  is  called  a  tie;  a  member  subject  to  compression 
or  crushing  is  called  a  strut. 

171.  For  the  purpose  of   dimensioning   the   members,  it  is 
necessary  to  know  the  stress  in  every  member.     The  following 


IO2 


STATICS. 


[171. 


example  illustrates  a  simple  method  for  finding  these  stresses 
when  the  external  forces  are  given. 

Let  the  frame-work  represented  in  Fig.  48  be  cut  in  two  along 
any  line  aft ;  the  portion  on  either  side  of  this  line  must  be  in 
equilibrium  under  the  action  of  its  external  forces  and  the 


Fig.  48. 


stresses  in  the  members  intersected  by  aft.  Thus,  in  the  figure, 
the  forces  A,  W^  Fv  F2,  Fz  form  a  system  in  equilibrium  ; 
hence,  the  sum  of  the  moments  of  these  forces  with  respect  to 
any  point  must  vanish. 

To  determine  'Fv  take  moments  about  the  intersection  of  F2 
and  FB  ;  thus  F%  and  FB  are  eliminated  from  the  equation  of 
moments,  and  F1  is  found.  Similarly  F2  is  obtained  by  taking 
moments  about  the  intersection  of  F3  and  Fr  The  arms  of  the 
moments  are  best  taken  from  a  correctly  drawn  diagram  of 
the  frame-work. 

If  only  two  members  be  intersected  by  aft,  the  origin  for 
the  moments  is  taken  first  on  one,  then  on  the  other,  of  the  two 
members  intersected. 

By  beginning  at  one  of  the  supports   and  taking  ^sections 


I73-] 


JOINTED    FRAMES. 


103 


through  the  successive  panels,  it  will  in  the  more  simple  cases 
be  possible  to  draw  the  line  «/3  so  as  to  intersect  not  more  than 
three  members  whose  stresses  are  unknown.  Thus  the  stresses 
in  all  the  members  can  be  determined. 

172.   Exercises. 

(i)  Find  the  stresses  in  the  braced  beam  A£  (Fig.  49),  carrying 
a  weight  of  5  tons  at  each  joint  of  the  upper  chord.  The  horizontal 
width  of  the  panels  is  10  ft.,  the  middle  vertical  is  8  ft. 


Fig.  49. 

(2)  In  Fig.  50,  the  dimensions  are  in  feet,  the  loads  in  tons.  After 
the  first  panel  the  sections  cannot  be  so  taken  as  to  intersect  not  more 
than  three  unknown  stresses.  But  the  girder  can  be  regarded  as 
obtained  by  the  superposition  of  two  girders  (each  carrying  half  the 
load) ,  in  one  of  which  the  diagonals  CF,  EH  are  wanting,  while  in  the 
other  DE,  FG  are  wanting.  Each  of  these  can  readily  be  computed. 


Fig.  50. 


4.     GRAPHICAL    METHODS. 

173.  The  graphical  method  explained  in  Art.  108  for  deter- 
mining the  resultant  of  a  system  of  parallel  forces  can  be 
extended  without  difficulty  to  the  general  case  of  a  plane  sys- 
tem of  forces.  The  only  difference  will  appear  in  the  form  of 
the  force  polygon,  which  for  parallel  forces  collapses  into 
a  straight  line,  while  in  the  general  case  it  is  an  ordinary 
(unclosed)  polygon  whose  closing  line  represents  the  resultant 


104  STATICS.  [174. 

in  magnitude  and  direction.  In  other  words,  when  the  forces 
are  not  parallel,  they  must  be  added  geometrically,  and  not 
algebraically. 

The  construction  of  the  funicular  polygon  and  its  properties 
are  the  same  as  for  parallel  forces. 

If  the  force  polygon  does  not  close,  the  given  system  is 
equivalent  to  a  single  resultant  represented  in  magnitude,  direc- 
tion, and  sense  by  the  closing  line  ;  its  position  is  obtained  from 
the  funicular  polygon  whose  initial  and  final  lines  must  inter- 
sect on  the  resultant. 

If,  however,  the  force  polygon  closes,  the  system  may  be 
equivalent  to  a  couple,  or  it  may  be  in  equilibrium.  The  dis- 
tinction between  these  two  cases  is  indicated  by  the  funicular 
polygon.  If  the  initial  and  final  lines  of  this  polygon  coincide, 
the  system  is  in  equilibrium  ;  if  they  are  merely  parallel,  these 
lines  are  the  directions  of  the  forces  of  the  couple  to  which  the 
whole  system  reduces.  The  magnitude  and  sense  of  the  forces 
of  the  resulting  couple  are  obtained  from  the  force  polygon. 

174.  Thus  it  follows  from  the  graphical  as  well  as  from  the 
analytical  method  that  a  plane  system  may  be  equivalent  to  a 
single  force,  or  to  a  couple,  or  to  zero.    In  the  first  case,  the  force 
polygon  does  not  close,  and  the  initial  and  final  sides  of  the 
funicular  polygon  intersect  at  a  finite  distance.     In  the  second 
case,  the  force  polygon  closes,  and  the  initial  and  final  lines  of 
the  funicular  polygon  are  parallel.     In  the  third  case,  the  force 
polygon  closes,  and  the  initial  and  final  sides  of  the  funicular 
polygon  coincide. 

The  graphical  conditions  of  equilibrium  of  a  plane  system  are, 
therefore,  two:  (i)  the  force  polygon  must  close;  (2)  the  funi- 
cular polygon  must  have  its  initial  and  final  sides  coincident. 

175.  To  every  vertex  of  the  force  polygon  corresponds  a  side 
of  the  funicular  polygon,  and  vice  versa.     The  force  polygon  is 
said  to  close  if  the  last  vertex  coincides  with  the  first ;  similarly, 
the  funicular  polygon  might  be  said  to  close  when  its  last  side 


GRAPHICAL   METHODS. 


105 


coincides  with  the  first.  With  this  convention,  we  may  say  that 
the  conditions  of  equilibrium  of  a  plane  system  require  the  closing 
of  both  the  force  polygon  and  the  funicular  polygon. 

176.  One  of  the  most  important  applications  of  the  graphical  methods 
is  found  in  the  determination  of  tJie  stresses  in  the  frame-works  used  for 
bridges,  roofs,  cranes,  etc.  The  following  example  will  illustrate  the 
method. 

Fig.  51  represents  the  skeleton  frame  of  a  roof  truss  subjected  to  the 


"loads" 


W  and  the 


reactions  of  the  supports  A,  B. 
The  members  of  the  frame  in 
connection  with  the  lines  of  ac- 
tion of  these  forces  (imagined  as 
drawn  from  infinity  up  to  the 
points  of  application)  divide  the 
whole  plane  into  a  number  of 
compartments  marked  in  the 
figure  by  the  letters  a,  b,  c,  d,  •••. 
The  external  forces  as  well  as 
the  members  of  the  frame  (or 
the  stresses  acting  along  them)  E 
can  thus  be  designated  by  the 
two  letters  of  the  two  portions 
of  the  plane  separated  by  the 
force  or  stress.  For  instance, 
the  reaction  A  is  denoted  ab, 
and  the  stresses  in  the  two  mem- 
bers concurring  at  A  are  be 
and  ca.  The  figure  just  de- 
scribed may  be  called  the  frame 
diagram ;  and  we  proceed  now 
to  construct  its  stress  diagram* 
Laying  off  on  a  vertical  line 
gj  =  \V\,  eg  =  IV^  be  =  IV$,  and 
bisecting  bj  at  a,  we  have  the 
polygon  of  the  external  forces  which  gives  the  reactions  A  =  ab,  B  =/#. 


Fig.  51. 


*The  student  is  advised  to  draw  the  stress   diagram  himself  step   by   step   as- 
indicated  in  the  text. 


106  STATICS.  [177. 

Next,  beginning  at  the  vertex  A  the  stresses  in  the  two  members 
intersecting  at  A  are  found  by  resolving  the  reaction  A  along  the  direc- 
tions of  these  members;  and  this  is  done  in  the  stress  diagram  by 
drawing  parallels  to  these  directions  through  the  points  a  and  b.  The 
intersection  is  denoted  by  c. 

177.  It  will  be  noticed  that  the  three  lines  meeting  at  A  have  corre- 
sponding to  them,  in  the   stress  diagram,  the  three  sides  ab,  be,  ca  of 
a  triangle.     The  force  A  =  ab  is  represented  by  ab  \  the  stress  in  the 
member  be  (i.e.  in  the  member  separating  the  compartments  b,  c  in 
the  frame  diagram)  is  represented  in  magnitude,  direction,  and  sense  by 
the  side  be  in  the  stress  diagram  ;  and  the  stress  in  the  member  ca  is 
given  by  the  side  ca  of  the  triangle  abc.     To  obtain  the  sense  of  each 
stress  correctly,  the  triangle  abc  in  the  stress  diagram  must  be  traversed 
in  the  sense  of  the  known  force  A  =  ab  •  this  shows  that  the  member  be 
is  compressed,  the  stress  at  A  acting  towards  A,  while  ca  is  subject  to 
tension. 

It  will  be  found  in  general  that  the  lines  of  the  stress  diagram  corre- 
sponding to  all  the  lines  meeting  at  any  one  vertex  of  the  frame  diagram 
form  a  closed  polygon.  The  reason  is  obvious  :  the  forces  at  the  vertex 
must  be  in  equilibrium. 

178.  To  continue  the  construction  of  the  stress  diagram,  we  pass  to 
.another  vertex  of  the  frame  diagram,  selecting  one  at  which  not  more 

than  two  stresses  are  unknown.  Thus  at  the  vertex  acd  the  stress  in  ac 
is  known,  being  represented  by  ac  in  the  stress  diagram.  Hence  drawing 
through  a  a  parallel  to  da,  through  c  a  parallel  to  cd,  we  find  the  point  d 
of  the  stress  diagram. 

The  vertex  dcbef  can  now  be  attacked  ;  dc,  cb,  be  are  already  drawn, 
and  it  only  remains  to  draw  ef  parallel  to  <f/"and  ^7"  parallel  to  df. 

The  rest  explains  itself.  Considerations  of  symmetry  are  frequently 
helpful  in  affording  checks. 

179.  Exercises. 

(1)  Check  the  computed  stresses  of  Exercises  (i)  and  (2),  Art.  172, 
by  constructing  the  stress  diagrams. 

(2)  Find  the  stresses  in  the  frame  (Fig.  52),  if  the  load  consists  of 
:  seven  equal  weights,  of  2  tons  each,  applied  at  the  joints  of  the  upper 

chord.     Owing  to  the  symmetry  of  the  figure,  it  is  sufficient  to  construct 


i8o.]  GRAPHICAL   METHODS.  IO/ 

the   stress  diagram  for  half  the  frame.     At  the  vertex  F,  a  difficulty 
arises,  there  being  apparently  three  members  whose  stresses  are  not 


I 


Fig.  52. 

known  from  the  previous  construction ;  but  on  account  of  the  symmetry 
with  respect  to  EF,  the  members  FG  and  FH  must  have  equal  stresses. 

180.  Shearing  Force  and  Bending  Moment.  Consider  a  hori- 
zontal beam  fixed  at  one  end  A  (Fig.  53),  and  acted  upon  at  the 
other  end  B  by  a  vertical  force  F.  If  the  beam  be  cut  at  any 
point  C  of  its  length,  and  the  equilibrium  of  the  portion  AC  be 
considered,  the  action  on  AC  of  the 
portion  removed  must  be  replaced 

by  its  equivalent.     Now  the  force      B          .C . 

F  at  B  is  equivalent,  by  Art.  136, 

to  an  equal  and  parallel  force  F  at        I 

C  in  connection  with  a  couple  whose  Fig.  53. 

moment  is  F-  BC. 

The  force  F  at  C  is  called  the  shearing  force  of  the  cross- 
section  C,  and  the  moment  F-BC  the  bending  moment  at  C. 
Both  are  of  great  importance  in  engineering,  as  their  combined 
effect  represents  what  must  be  overcome  by  the  resistance  of 
the  material  of  the  beam,  i.e.  by  the  internal  forces  holding 
together  its  fibres. 

These  definitions  are  readily  generalised.  Let  any  beam  or 
girder,  supported  in  any  manner,  and  acted  upon  by  any  number 
of  vertical  forces,  be  divided  by  a  vertical  cross-section  into  two 
portions  A  and  B.  For  the  portion  A  the  shearing  force  at  the 
cross-section  is  the  sum  of  all  the  external  forces  acting  on  B ; 
and  the  bending  moment  is  the  sum  of  the  moments  of  all  these 
forces  with  respect  to  some  point  in  the  cross-section. 


1 08 


STATICS. 


[181. 


181.  According  to  its  definition  the  bending  moment  of  a 
beam  at  any  cross-section  is  found  by  adding  the  moments,  with 
respect  to  the  cross-section,  of  all  the  external  forces  on  one 
side  of  the  section. 

Graphically,  the  bending  moment  is  readily  derived  from  the 
funicular  polygon.  Thus  in  Fig.  54,  for  the  cross-section  a/3, 
the  resultant  of  the  forces  on  the  left  is  R'  =  A  —  IV1  —  W2  =  o  3  in 


Fig.  54. 

the  force  polygon.  Its  position  is  found  by  bringing  to  inter- 
section the  two  sides  A'B1  and  II  III  of  the  funicular  polygon 
met  by  the  section  a/3.  For  the  funicular  polygon  resolves  A 
along  A*B'  and  A'l,  Wl  along  I  A'  and  I  II,  IV2  along  II  I  and 
II  III.  The  components  falling  into  the  same  line  being  equal 
and  opposite  (as  appears  from  the  force  polygon),  the  forces  A, 
Wlt  JV2  are  together  equivalent  to  the  components  along  A'B* 
and  II  III ;  their  resultant  R'  must  therefore  pass  through  the 
intersection  5  of  these  lines. 


183.]  FRICTION. 

Now  if  p  be  the  horizontal  distance  of  the  point  5  from  aft, 
the  bending  moment  at  a/3  is  R1  -/  =  o  3  -p.  If  a/3  intersect  A'B' 
in  P,  II  III  in  Q,  the  triangles  SPQ  and  #03  are  similar,  so 
that  their  altitudes  p  and  H  are  as  the  homologous  sides  PQ 
and  o  3  ;  hence 


and  the  value  of  the  bending  moment  is  H  •  PQ.  As  H  is 
constant,  we  find  that  the  bending  moment  is  proportional  to  the 
vertical  height,  or  ordinate,  of  the  funicular  polygon. 

5*    FRICTION. 

182.  The  reaction  between  two  surfaces  in  contact  has  so  far 
been  regarded  as  directed  along  the  common   normal  of  the 
surfaces.     This  is  true  when  the  surfaces  are  perfectly  smooth. 

The  surfaces  of  physical  bodies  are  rough,  i.e.  they  present 
small  elevations  and  depressions ;  when  two  such  surfaces  are 
"in  contact"  the  projections  of  one  will  more  or  less  enter  into 
depressions  of  the  other ;  the  greater  the  normal  pressure 
between  the  surfaces,  the  more  will  this  be  the  case.  Hence 
when  a  tangential  force  acting  on  one  of  the  bodies  tends  to 
slide  its  surface  over  that  of  the  other  body,  a  resistance  will  be 
developed  whose  magnitude  must  depend  on  the  roughness  of 
the  surfaces  and  on  the  normal  pressure  between  them.  This 
resistance  is  called  the  force  of  friction. 

The  study  of  friction  belongs  properly  to  applied  mechanics, 
and  will  here  only  be  touched  upon  very  briefly. 

183.  Imagine  a  body  resting  with  a  plane  surface  on  a  hori- 
zontal plane.     Let  a  small  horizontal  force  P  be  applied  at  its 
centroid  (which  is  supposed  to  be  situated  so  low  that  the  body 
is  not  overturned),  and  let  the  force  P  be  gradually  increased 
until  motion  ensues.     The  value  of  P  when  motion  just  begins 
is  equal  and  opposite  to  the  frictional  resistance  F  between  the 


1 10  STATICS.  [184. 

surfaces  at  this  moment,  and  this  resistance  is  called  the  limiting 
static  friction. 

Careful  experiments  have  shown  this  force  to  be  subject  to 
the  following  laws  : 

(i)  The  magnitude  of  the  limiting  friction  F  bears  a  constant 
ratio  to  the  normal  pressure  N  between  the  surfaces  in  contact ; 
that  is 


where  //,  is  a  constant  depending  on  the  condition  and  nature  of 
the  surfaces  in  contact.  This  constant  which  must  be  deter- 
mined experimentally  for  different  substances  and  surface 
conditions  is  called  the  coefficient  of  static  friction.  It  is  in 
general  a  proper  fraction  ;  for  perfectly  smooth  surfaces  /JL  =  O. 

(2)  For  a  given  normal  pressure  the  limiting  static  friction, 
and  hence  the  coefficient  of  static  friction,  is  independent  of  the 
area  of  contact, 

184.  The  frictional  resistance  between  two  surfaces  in  rela- 
tive motion  is  called  kinetic  friction.     It  is  subject,  in  addition 
to  the  two  laws  just  mentioned,  to  the  third  law : 

(3)  Kinetic  friction  is  independent  of  the  velocities  of  the  bodies 
in  contact. 

The  coefficient  of  static  friction  is  generally  slightly  greater 
than  that  of  kinetic  friction. 

It  must  not  be  forgotten  that  these  so-called  laws  of  friction 
are  experimental  laws,  and  therefore  true  only  approximately, 
and  within  the  limits  of  the  experiments  from  which  they  were 
deduced.  When  the  relative  velocity  of  the  surfaces  in  contact 
is  very  high,  and  when,  as  is  usually  the  case  in  machinery, 
lubricating  material  is  introduced  between  the  two  surfaces,  th 
frictional  resistance  is  found  to  depend  on  a  number  of  other 
circumstances,  such  as  the  temperature,  the  form  of  the  sur^ 
faces,  the  velocity,  the  nature  of  the  lubricator,  etc. 

185.  Consider  again   a   body  resting   on  a  horizontal  plane 
(Fig.  55),  and  acted  upon  by  a  horizontal  force  PJust  large- 


i 


i86.] 


FRICTION. 


I II 


enough  to  equal  the  limiting  friction  F.  The  normal  reaction 
N  of  the  plane  is  equal  and  opposite  to  the  weight  W.  The 
body  is  thus  in  equilibrium  un- 
der the  action  of  the  two  jDairs  of 
equal  and  opposite  forces ;  but 
motion  will  ensue  as  soon  as  P 
is  increased.  If  P  be  decreased, 
.Fwill  decrease  at  the  same  rate, 
so  that  the  equilibrium  remains  |W 

undisturbed.  Fi£-  55- 

The  force  of  friction  Fcan  be  combined  with  the  normal  reac- 
tion Nto  form  a  resultant, 


which  represents  the  total  reaction  of  the  horizontal  plane. 

If  $  be  the  angle  between  N  and  R  when  F  has  its  limiting 
value  F=pN(Art.  183),  we  have,  since  tan<£  =  F/Ar, 

tan  cf)  =  fju. 

The  angle  <£  thus  gives  a  kind  of  graphical  representation  for 
the  coefficient  of  friction  p ;  it  is  called  the  angle  of  friction. 

186.    If  the  plane  be  not  horizontal,  but  inclined  to  the  hori- 

,<•*  zon  at  an  angle  6,  the  weight  W 
>/  of  the  body  (regarded  as  a  particle) 
resting  on  the  plane  can  be  re- 
solved into  a  component  J^Fsin# 
along  the  plane,  and  a  component 
W  cos  0  perpendicular  to  it  (Fig. 
56).  Hence,  if  no  other  forces 
act  on  the  body  it  will  be  in  equi- 
librium, provided  the  component 
Ws'mO  be  not  greater  than  the 
The  limiting  condition  of  equi- 


Fig.  56. 

limiting  friction  F=/J,  Wcos  6. 
librium  is,  therefore, 

fj,  IV  cos  0  =  Wsin  0, 


or    //,  =  tan  0  ; 


112  STATICS.  [187. 

in  other  words,  if  the  angle  9  be  gradually  increased,  the 
body  will  not  slide  down  the  plane  until  6  >  <£.  This  fur- 
nishes an  experimental  method  of  determining  the  angle  of 
friction  $,  which  on  this  account  is  sometimes  called  the  angle 
of  repose. 

187.  A  particle  P  (Fig.  57)  will  be  in  equilibrium  on  any 
rough  surface,  if  the  total  reaction  of  the  surface,  i.e.  the  result- 

/         ant  R  of  the  normal  reaction 
N  and  the  friction  F,  is  equal 
and  opposite  to  the  resultant 
/      R'  of  all  the  other  forces  act- 
ing on  the  particle. 

The   limiting   value   of   the 
angle  between  N  and  R  is  </>, 
so  that  the  particle  can  be  in 
57<  equilibrium  only  if  the  result- 

ant R'  makes  with  the  normal  an  angle  <<£.     Hence,  if  about 
the   normal  PN  as    axis,   and  with  P  as   vertex,   a   cone   be 
described  whose  vertical  angle  is   2$,   the  condition  of  equi- 
librium is  that  R'  must  lie  within  this  cone. 
The  cone  is  called  the  cone  of  friction. 

188.  Exercises. 

(1)  A  particle  of  weight  W"\s  in  equilibrium  on  a  rough  plane  inclined 
to  the  horizon  at  an  angle  6,  under  the  action  of  a  force  P  parallel  to 
the  plane  along  its  greatest  slope.     Determine  P:   (a)  when  6  >  <£,  (^) 
when  0  =  <f>,  (c)  when  6  <  <£,  <£  =  tan"1/*  being  the  angle  of  friction. 

(2)  Determine  the  tractive  force  required  to  haul  a  train  of  100  tons 
with  constant  velocity  up  a  grade  of  2.5  per  cent  if  the  coefficient  ol 
friction  is  1/200. 

(3)  A  weight  W  is  to  be  hauled  along  a  horizontal  plane,  the  coeffi- 
cient of  friction  being  /*  =  tan  </>.     Determine  the  required  tractive  force 
P  if  it  is  to  act  at  an  inclination  a.  to  the  horizon,  and  show  that  this 
force  is  least  when  «=</>. 


189-]  FRICTION.  II3 

(4)  A  particle  of  weight  Wis  kept  in  equilibrium  on  a  plane  inclined 
at  an  angle  6  to  the  horizon  by  a  force  P  making  an  angle  a  with  the 
line  of  greatest  slope  (in  the  vertical  plane  at  right  angles  to  the  intersec- 
tion of  the  inclined  plane  with  the  horizon).     Find  the  conditions  of 
equilibrium  when  the  particle  is  on  the  point  of  moving  (a)  down  the 
plane,  (^)  up  the  plane. 

(5)  A  homogeneous  straight  rod  AB  =  2/  of  weight  Crests  with  one 
end  A  on  the  horizontal  floor,  with  the  other  end  B  against  a  vertical  wall 
whose  plane  is  at  right  angles  to  the  vertical  plane  of  the  rod.     If  there 
be  friction  of  angle  <£  at  both  ends,  determine  the  limiting  position  of 
equilibrium. 

(6)  Two  particles  whose  weights  are  W,  W  are  in  equilibrium  on  an 
inclined  plane,  being  connected  by  a  string  directed  along  the  line  of 
greatest  slope.     If  the  coefficients  of  friction  are  /*,  /x',  determine  the 
inclination  of  the  plane. 

189.  The  idea  of  the  angle  of  friction  suggests  a  graphical  method  for 
problems  on  equilibrium  with  friction. 

The  case  of  a  rod  resting  on  two  inclined  planes,  Art.  150,  Fig.  41, 
may  serve  as  an  example.  If  the  intersection  E  of  the  normal  reactions 
A  and  B  lies  on  the  vertical  through  D,  the  rod  will  be  in  equilibrium 
whether  there  be  friction  at  A  and  B  or  not.  When  this  condition  is 
not  fulfilled,  the  rod  may  still  be  in  equilibrium  if  there  be  sufficient  fric- 
tion between  the  ends  of  the  rod  and  the  supporting  planes. 

Let  p.  =  tan  $  be  the  coefficient  of  friction  on  the  plane  CA,  /x'  =  tan<£f 
that  on  CB ;  then  the  total  reactions  at  A  and  B  will,  by  Art.  185, 
make  angles  not  greater  than  <£  and  <f>',  respectively,  with  the  normals  to 
the  planes.  Hence  the  two  limiting  positions  of  equilibrium  for  the 
weight  W,  in  a  given  position  of  the  rod,  can  be  found  by  bringing  the 
lines  of  these  total  reactions  to  intersection  ;  the  limiting  position  of  W 
is  the  vertical  through  this  intersection.  Thus,  to  prevent  the  rod  from 
sliding  up  the  plane  CA  and  down  the  plane  CB.,  the  friction  angles  <£, 
<£'  must  be  applied  in  the  negative  sense  (clockwise)  to  the  normals  at 
A  and  B ;  this  gives  one  limiting  position  D1  for  the  point  D.  The 
other  position  D"  is  found  by  applying  the  friction  angles  in  the  positive 
sense.  Equilibrium  will  therefore  subsist  if  the  weight  be  placed  any- 
where between  D1  and  D". 


PART   II — 8 


STATICS. 


[190. 


The  construction  is  somewhat  simplified  when  <£  =  <£'  since  then  the 
intersections  of  the  total  reactions  lie  on  the  circle  described  about 
ABC  (Fig.  58). 


W 


Fig.  58. 


190.   As  another  example  consider  the  ordinary  jack  intended  to  raise 
an  eccentric  load  JF  acting  vertically  downwards  through  A  (Fig.  59)  by 

a  force  P  passing  vertically  upwards  through 
the  pitch-line  B  of  the  rack.  Near  C  and  D 
the  rack  is  pressed  against  the  casing.  The 
directions  of  the  total  reactions  C,  D  at 
these  points  are  found  by  applying  the 
friction  angle  to  the  normals. 

The  four  forces  W,  P,  C,  D  can  be  in 
equilibrium  only  if  the  resultant  of  W  and 
D  is  equal  and  opposite  to  the  resultant 
of  P  and  C ;  hence,  if  E  be  the  intersec- 
tion of  W  and  D,  F  that  of  .P  and  C,  each 
of  these  resultants  must  act  along  EF. 

If    the   load    W    be   known,   the   other 
forces  can  now  be  found  by  constructing 
the    force    polygon.       Draw    i  2  =  W   in 
position  (i.e.  through  A)  •  draw  2  3  paral- 
Fig>  59<  lei  to  C;  41  parallel  to  D]   and  through 

the  intersection  4  of  4  i  with  EF  draw  the  vertical  3  4  to  the  intersec- 
tion 3  with  2  3. 


/ 


192.] 


FRICTION. 


191.  Journal  Friction.    A  journal,  or  trunnion,  is  the  cylindrical  end 
of  a  horizontal  shaft,  by  means  of  which  the  shaft  is  supported  in  its 
bearing.    The  larger  circle  in  Fig.  60  represents  a  cross-section  of  the 
journal  at  right  angles  to  the  axis  of  the  shaft. 

The  shaft  and  journal  may  be  regarded  as  rotating  uniformly  about 
their  common  horizontal  axis  under  the  action  of  a  driving  force  whose 
moment  with  respect  to  a  point  O  on  the  axis  would  have  to  be  exactly 
equal  and  opposite  to  that  of  the  resistance,  or  load,  if  there  were  no 
journal  friction.  For,  in  this  case,  the  reaction  of  the  bearing  to  the 
weight  WQ{  the  shaft  would  act  vertically 
upwards  through  the  axis  of  the  shaft,-  so 
that  its  moment  would  be  zero. 

The  existence  of  friction  at  the  place 
of  contact  A  between  journal  and  bearing 
requires  an  increase  of  the  driving  force, 
which  may  be  regarded  as  a  small  tan- 
gential force  P  applied  at  any  point  B, 
such  that  its  moment  P-  OB  equals  the 
moment  about  O  of  the  frictional  resist- 
ance at  A. 

192.  Let  C  be  the  intersection  of  the 
direction  of  this  force  /'with  the  vertical 

through  O  and  A,  which  is  the  line   of  Pig   50. 

action   of  the   weight    W  of  the  shaft. 

The  resultant  of  P  and  W  passes  through  C,  and  intersects  the  circum- 
ference of  the  journal  in  a  point  D  near  A ;  the  total  reaction  of  the 
bearing  is  equal  and  opposite  to  this  resultant.  As  the  total  reaction 
must  make  an  angle  equal  to  the  angle  of  friction  <£  with  the  normal  at 
D  which  passes  through  the  centre  O,  we  have  for  the  perpendicular 
OE  dropped  from  O  on  CD, 

OE  =  p  =  r  sin  <f>, 

where  r  is  the  radius  of  the  journal.  A  circle  described  about  O,  with 
p  as  radius,  has  the  total  reaction  of  the  bearing  as  a  tangent.  This 
circle  is  called  the  friction  circle.  As  <J>  is  generally  very  small  in  the 
case  of  journal  friction,  //,  =  tan<£  can  be  substituted  for  sin<£,  and  we 
have  for  the  radius  p  of  the  friction  circle 


As  soon  as  any  one  point  is  known  through  which  the  total  reaction 


Il6  STATICS.  [193. 

must  pass  (as  the  point  C  in  Fig.  60),  its  direction  is  found  by  drawing 
through  this  point  a  tangent  to  the  friction  circle. 

193.  If  the   shaft   revolved   in  the   opposite   sense,   i.e.   clockwise 
(instead  of  counter-clockwise,  as  assumed  in  Fig.   60),  the  tangent  to 
the  friction  circle  would  have  to  be  drawn  through  C  on  the  other  side 
of  the  friction  circle. 

In  the  case  of  axle-friction,  i.e.  when  the  journal,  or  axle,  is  fixed, 
while  the  bearing,  or  hub,  revolves  about  it,  the  same  considerations 
would  apply,  except  that  the  point  of  application  of  the  total  reaction 
would  now  be  at  the  top,  at  D',  instead  of  D. 

194.  Pin-friction,  as  it  occurs  in  link-work  and  jointed  frames  that 
are  not  absolutely  stiff,  is  not  different  from  journal  friction  or  axle- 
friction,  and  can  be  treated  in  the  same  way.     Thus,  a  link  connected 
to  other  parts  of  a  machine  by  means  of  a  pin  at  each  end  would  trans- 
mit the  force  along  the  line  joining  the  centres  of  the  pins  if  there  were 
no  friction.     To  take  account  of  pin-friction,  we  have  only  to  draw  the 
friction  circles  about  the  centre  of  each  pin;  the  direction  in  which 
the  force  is  transmitted  by  the  link  is  tangent  to  both  these  circles. 

Which  one  of  the  four  common  tangents  represents  this  direction  must 
be  decided  in  each  particular  case  by  considering  that  the  reaction 
exerted  by  one  link  on  another  connected  with  it  by  a  pin  is  in  the 
direction  of  the  motion  of  the  former  relative  to  the  other.  Thus  if 
the  link  AB  (Fig.  61)  be  subject  to  tension,  and  its  motion  relative  to 


Fig.  61. 

the  adjoining  links  at  A  and  B  be  as  indicated  by  the  arrows  in  the  figure, 
the  contact  between  the  link  and  pin  will  be  on  the  outside  both  at  A  and 
at  B ;  the  friction  is,  therefore,  directed  downwards  at  A  and  upwards  at 
B,  and  the  line  PQ  along  which  the  force  is  transmitted  touches  the 
friction  circle  at  A  below,  at  B  above. 

If  the  link  were  under  compression,  with  the  same  relative  motions, 
the  line  of  force  would  have  the  direction  P'  Q'. 


FRICTION. 


117 


195.  The  simplest  case  of  pivot  friction  is  that  of  a  vertical  shaft  of 
weight  W  resting  with  its  circular  end  on  a  plane  horizontal  support. 
If  a  be  the  radius  of  the  end  of  the  shaft,  the  pressure  per  unit  of  area 

is  W/TTO?,  and  the  pressure  on  a  polar  element  of  area  is  — -  •  rdrdO. 

W  'Ira 

The  friction  at  this  element,  ^ — -  -  rdrdQ,  is  directed  along  the  tangent 

to  the  circle  of  radius  r ;  its  moment  with  respect  to  the  centre  O  of  the 

circle  is  therefore  ^-^rzdrdO.     Hence  the  whole   moment   of  friction 
about  O  is  va 


ira' 


This  may  be  regarded  as  the  moment  of  a  force  /x  W  applied  at  a 
distance  \a  from  the  centre. 

196.  Belt-friction,  A  belt  running  over  two  pulleys  and  stretched  so 
tight  as  to  prevent  slipping  is  a  common  means  of  transferring  the 
rotary  motion  about  the  axis  of  one  pulley,  say  At  to  the  axis  of  the 
other  pulley  B ;  A  is  called  the  driver,  B  is  the  driven  pulley.  We 
assume  the  axes  parallel  and  the  rotation  counter-clockwise. 

When  the  pulleys  are  at  rest  the  tension  in  CE  (Fig.  62)  is  of  course 


Fig.  62. 

equal  to  the  tension  in  DF.  But  if  the  pulley  A  be  set  in  motion,  say 
by  a  tangential  driving  force  P  acting  at  a  lever-arm  /,  while  the  pulley 
B  experiences  a  resistance  Q  whose  arm  is  q,  the  tension  in  CE  will 
increase  to  a  certain  value  Tlt  and  the  tension  in  DF  will  decrease  to 
a  value  T2  until  the  difference  7J— 7"2  is  sufficient  to  overcome  the 
resistance  Q.  This  difference  is  due  to  the  friction  along  the  surface 
CGD.  If  the  resistance  Q  be  too  great  this  friction  might  not  be  suffi- 
cient, and  slipping  of  the  belt  on  the  driver  would  occur. 


STATICS. 


[J97- 


197.  Let  us  try  to  determine  the  condition  which  71  and  T2  must 
satisfy  to  prevent  slipping.  To  do  this  we  determine  the  equilibrium  of 
the  belt  at  the  moment  when  slipping  is  just  on  the  point  of  taking 
place. 

The  tension  of  the  belt  decreases  gradually  along  the  arc  CGD  from 
the  value  71  at  C  to  the  value  Ts  at  D.  Let  it  be  T  at  the  point  P  and 
T+  dT  'at  the  near  point  P'  (Fig.  63).  The  portion  PP'  of  the  belt  is 
in  equilibrium  under  the  action  of  the  forces  T,  T  -\-  dTand  the  reaction 

dR  of  the  pulley  ;  hence  dR  must  pass 
through  the  intersection  of  T  and 
T-\-  dT  and  must  make  with  the  radius 
APan  angle  equal  to  the  friction  angle  <£. 
Resolving  these  forces  along  T  and  at 
right  angles  to  it,  we  have,  if  %.PAP'=dO, 

T+  dR  sin  <£  =  T+  dT, 


or          dR  sin  <f>  =  dT, 
dR  cos  <£  =  TdO  ; 

T+dT  hence,  dividing, 

i  dT 


Fig.  63. 


Putting  /u,  for  tan  <£  and  integrating  over  the  whole  arc  of  contact,  we 
find,  if  0  be  the  angle  of  this  arc, 

log  71  -log  T2  =  n$, 


or 


T 


For  the  common  system  of  logarithms  this  becomes 

log— *  =  0.4343 /*0, 
where  0  must  be  expressed  in  circular  measure. 

198.  Rolling  Friction.  The  resistance  offered  by  a  surface  to 
tbe  rolling  of  another  surface  over  it  is  of  a  somewhat  different 
nature  from  that  of  ordinary  or  sliding  friction.  In  sliding  fric- 
tion, the  same  point  or  surface  area  of  one  body  comes  in 


199-]  FRICTION.  IIC> 

•contact  with  different  points  or  areas  of  the  other.  In  the  case 
•of  rolling  friction,  the  points  that  come  successively  in  contact 
.are  different  for  both  bodies. 

Let  us  examine  the  simplest  case,  viz.  that  of  a  cylinder  roll- 
ing over  a  horizontal  plane.  If  both  cylinder  and  plane  were 
perfectly  rigid,  there  could  be  no  resistance  to  rolling.  This 


Fig.  64. 

resistance  is  due  to  the  compression  both  of  the  lower  part  of 
the  cylinder  and  of  the  plane.  Experiments  made  with  a  heavy 
roller  on  india-rubber  have  shown  that  the  supporting  surface 
when  elastic  is  not  only  compressed  under  the  roller  but  bulges 
•out  in  front  and  behind,  as  indicated  in  Fig.  64.  Thus,  the  area 
of  contact  is  considerably  increased,  and  as  the  roller  advances, 
the  portion  AB  of  its  surface  rubs  over  the  surface  of  the  sup- 
port, while  the  elastic  material  of  the  support  in  trying  to  regain 
its  horizontal  surface  causes  friction  over  the  area  B1A  also. 

The  experiments  indicate  for  the  value  Fof.  rolling  friction  an 
expression  of  the  form 


where  W  is  the  weight,  r  the  radius  of  the  cylinder,  and  ^  a 
constant  depending  on  the  nature  of  the  materials  in  contact. 
For  hard  surfaces,  this  constant  of  rolling  friction  p'  is  very 
much  smaller  than  the  constant  of  sliding  friction  /z. 

199.    On  the  subject  of  plane  statics  the  student  may  consult  in  par- 
ticular the  recent  work  :    E.  J.  ROUTH,  A  treatise  on  analytical  statics^ 


120  STATICS.  [199. 

with  numerous  examples,  Vol.  I.,  Cambridge,  University  Press,  1891  ; 
also  G.  M.  MINCHIN,  A  treatise  on  statics  with  applications  to  physics, 
Vol.  I.,  3d  ed.,  Oxford,  Clarendon  Press,  1884  ;  I.  TODHUNTER,  Analyti- 
cal statics,  5th  ed.  by  J.  D.  Everett,  London,  Macmillan,  1887  ;  B.  PRICE, 
Infinitesimal  calculus,  Vol.  III. :  Statics  and  dynamics  of  material  par- 
ticles, 2d  ed.,  Oxford,  Clarendon  Press,  1868.  For  problems,  see  also 
•W.  WALTON,  Collection  of  problems  in  illustration  of  the  principles  of 
elementary  mechanics,  2d  ed.,  Cambridge,  Deighton,  1880. 

Numerous  applications  to  civil  and  mechanical  engineering  will  be 
found  in  J.  H.  COTTERILL,  Applied  mechanics,  London,  Macmillan,  1884  ; 
W.  J.  M.  RANKINE,  A  manual  of  applied  mechanics,  9th  ed.  by  E.  F. 
Bamber,  London,  Griffin,  1877;  A.  RITTER,  Lehrbuch  der  technischen 
Mechanik,  5th  ed.,  Leipzig,  Baumgartner,  1884 ;  J.  WEISBACH,  Mechan- 
ics of  engineering,  Vol.  I.,  translated  by  E.  B.  COXE,  New  York,  Van 
Nostrand,  1875  ;  and  in  works  on  graphical  statics. 

On  friction,  see  in  particular  :  G.  HERRMANN,  The  graphical  statics  of 
mechanism,  translated  by  A.  P.  Smith,  2d  ed.,  New  York,  Van  Nostrand, 
1892  ;  R.  H.  THURSTON,  Treatise  on  friction  and  lost  work  in  machinery 
and  mill  work,  New  York,  Wiley,  1885  ;  J.  H.  JELLETT,  The  theory  of 
friction,  Dublin,  Hodges,  1872. 


200.] 


CONDITIONS   OF   EQUILIBRIUM. 


121 


VI.    Solid  Statics. 

I.     THE    CONDITIONS    OF    EQUILIBRIUM. 

200.  The  equilibrium  of  a  rigid  body  in  the  most  general 
case,  that  is,  when  acted  upon  by  any  number  of  forces  F  in  a 
space  of  three  dimensions,  can  be  investigated  in  a  manner 
similar  to  that  adopted  for  the  plane  system  in  Art.  139. 

Selecting  as  origin  any  point  O  rigidly  connected  with  the 
body,  let  two  equal  and  opposite  forces  F,  —  F  be  applied  at  O, 
for  every  one  of  the  given  forces  F  (Fig.  65).  The  effect  of  the 
given  system  of  forces  on 
the  body  is  not  changed  by 
the  introduction  of  these 
forces  at  O.  But  we  may 
now  regard  the  given  force 
F  acting  at  its  point  of 
application  P  as  replaced  by 
the  equal  and  parallel  force 
Fat  O,  in  combination  with 
the  couple  formed  by  the  original  force  F  at  P  and  the  force  —  F 
at  O.  All  the  forces  of  the  given  system  are  thus  transferred 
to  a  common  point  of  application  O,  and  can  therefore  be  com- 
pounded into  a  single  resultant  R,  passing  through  O  and 
represented  in  magnitude  and  direction  by  the  geometric  sum  of 
the  forces.  In  addition  to  this  resultant  R,  we  obtain  as  many 
couples  (Ft  —F)  as  there  were  forces  given  ;  and  their  resultant 
is  found  by  geometrically  adding  the  vectors  of  the  couples 
(Art.  134). 

Thus  the  given  system  of  forces  is  seen  to  be  equivalent  to  a 
resultant  R  in  combination  with  a  couple  whose  vector  we  shall 
call  H\  in  other  words,  it  has  been  proved  that  any  system  of 
forces  acting  on  a  rigid  body  can  be  reduced  to  a  single  resultant 
force  in  combination  isjith  a  single  resultant  couple. 


Fig.  65. 


122 


STATICS. 


201.    A  further   reduction  is  in  general  not   possible. 
general  conditions  of  equilibrium  are,  therefore, 


[201. 

The 


202.  Under  special  conditions  it  may  of  course  happen  that 
R  is  perpendicular  to  the  vector  H.     In  this  case  R  and  H  com- 
bine to  a  single  force  R  (Art.  135),  and  if  the  origin  be  taken  on 
the  line  of  this  force,  the  whole  system    reduces   to   a   single 
resultant. 

203.  It  is  to  be  noticed  that  in  the  general  reduction  of  forces 
(Art.  200),  the  magnitude,  direction,  and  sense  of  the  resultant 
force  R  are  entirely  independent  of  the  position  of  the  origin  O, 
the  resultant  being  simply  the  geometric  sum  of  all  the  given 
forces.     The  resultant  couple  H,  on  the  other   hand,   will   in 
general  differ  according  to  the  origin  selected. 

To  investigate  this  dependence,  let  Rt  H  (Fig.  66)  be  the  ele- 
ments of  reduction  for  the  origin  O\  i.e.  let  R  be  the  resultant, 

H  the  vector  of  the  resulting  couple  of 
a  given  system  of  forces  when  O  is 
selected  as  origin.  To  find  the  ele- 
ments of  reduction  of  the  same  system 
of  forces  when  some  other  point  O'  is 
taken  as  origin,  it  is  only  necessary  to 
apply  at  O'  two  equal  and  opposite 
forces  R,  —R,  each  equal  and  parallel 
to  the  original  resultant  R.  The  given 
system  of  forces  being  equivalent  to 
R  and  H  at  O  will  also  be  equivalent 
to  the  resultant  R  at  O1,  the  couple 
whose  vector  is  H  (which  may  be 
drawn  through  O'  without  changing  its  effect),  and  the  couple 
formed  by  R  at  O  and  -R  at  O'.  If  /  be  the  line  of  R  through 
O,  V  the  line  of  R  through  O',  and  r  the  distance  of  these 
parallels,  the  moment  of  the  latter  couple  is  Rr  and  its  vector  is 
at  right  angles  to  the  plane  (/,  O').  Combining  the  vectors  H 


-R 


I1 


Fig.   66. 


206.] 


CONDITIONS   OF   EQUILIBRIUM. 


123 


and  Rr  into  a  resultant  vector  H1  by  geometric  addition,  we 
have  found  the  elements  of  reduction  R,  H'  for  the  origin  O'. 

204.  If  the  new  origin  O'  had  been  selected  on  the  line  /  of 
the  original  resultant,  no  new  couple  (R,  r)  would  have  been 
introduced,  and  H  would  not  have  been  changed.     But  when- 
ever the  line  of  action  /  of  the  resultant  is  changed,  the  vector 
of  the  resultant  couple  H  is  changed. 

By  increasing  the  distance  r  between  /  and  /'  the  moment  Rr 
of  the  additional  couple  is  increased.  The  effect  of  combining 
this  additional  couple  Rr  with  H  is,  in  general,  to  vary  both  the 
magnitude  of  the  resulting  couple  H'  and  the  angle  <f>  it  makes 
with  the  direction  of  the  resultant  R.  It  can  be  shown  that  the 
line  /'  of  the  new  resultant  can  always  be  selected  so  as  to 
reduce  the  angle  <f>  to  zero.  The  line  /0  for  which  $  =  o,  i.e.  for 
which  the  vector  H  of  the  resultant  couple  is  parallel  to  the 
resultant  force  R,  is  called  the  central  axis  of  the  given  system 
of  forces.  We  proceed  to  show  how  it  can  be  found. 

205.  Let  the  vector  H  be  resolved  at  O  into  a  component 


J~fQ  =  H  cos  $  along  /,  and  a  component 
angles  to  /(Fig.  67).  In  the  plane  pass- 
ing through  /  at  right  angles  to  Hly  it 
is  always  possible  to  find  a  line  /0  par- 
allel to  /  at  a  distance  rQ  from  /,  such 
as  to  make  RrQ  =  —  Hv 

The  line  /0  so  determined  is  the  cen- 
tral axis.  For,  if  this  line  be  taken  as 
the  line  of  the  resultant  R,  the  addi- 
tional couple  RrQ  destroys  the  compo- 
nent ffv  so  that  the  resulting  couple 
HQ  has  its  vector  parallel  to  R. 

206.  As  the  direction  of  the  vector 
H  is  always  changed  in  passing  from 


at   right 


Fig.  67. 


line  to  line,  there  can  be  but  one  central  axis  for  a  given  system 
of  forces. 


124 


STATICS. 


[207. 


It  appears  from  the  construction  of  the  central  axis  given  in 
Art.  205,  that  the  vector  of  the  resulting  couple  for  this  axis  /0 
is  HQ  =  HCOS$ ;  it  is,  therefore,  less  than  for  any  other  line. 

It  is  instructive  to  observe  how  the  vector  H  increases  and 
changes  its  direction  as  we  pass  from  the  central  axis  /0  to  any 
parallel  line  /. 

The  transformation  from  /0  to  /  requires  the  introduction  of  a 
couple  (R,  rQ)  whose  vector  RrQ  (Fig.  68)  is  at  right  angles  to 
the  plane  (/0,  /)  and  combines  with  //"0  to  form  the  resulting 

couple  H  for  /.  As  the  distance  rQ 
of  /  from  /0  is  increased,  both  the 
magnitude  of  H  and  the  angle  </>  it 
makes  with  /  increase  until,  for  an 
infinite  rQ,  the  angle  $  becomes  a 
right  angle. 


-R 
I- 


207.  It  is  evident  that  since  i 
=  ff  cos  (f>,  the  product  RH  cos  0  is  a 
constant  quantity  for  a  given  system 
of  forces.  It  has  been  called  the 
invariant  of  the  system. 

If  the  elements  of  reduction  for  the 
central  axis  (R,  HQ)  be  given,  those 
for  any  parallel  line  /  at  the  distance  rQ  from  the  central  axis  are 
determined  by  the  equations 


Fig.  68. 


208.  To  sum  up  the  results  of  the  preceding  articles,  it  has 
been  shown  that  any  system  of  forces  acting  on  a  rigid  body  can 
be  reduced,  in  an  infinite  mimber  of  ways,  to  a  resultant  R  in 
combination  with  a  couple  H.  For  all  these  reductions  the  mag- 
nitude, direction,  and  sense  of  the  resultant  R  are  the  same,  but 
the  vector  H  of  the  couple  changes  according  to  the  position 
assumed  for  the  line  of  R.  There  is  one,  and  only  one,  position 
of  R,  called  the  central  axis  of  the  system,  for  which  the  vector 


211.] 


CONDITIONS   OF   EQUILIBRIUM. 


125 


H  is  parallel  to  R,  and  has  at  the  same  time  its  least  value,  HQ ; 
this  value  H§  is  equal  to  the  projection  of  any  other  vector  H 
on  the  direction  of  the  resultant  R. 

209.  While,  in  general,  a  system  of  forces  cannot  be  reduced 
to  a  single  resultant,  it  can  always  be  reduced  to  two  non-inter- 
secting forces.  This  easily  fol- 
lows by  considering  the  system 
reduced  to  its  resultant  R  and 
resulting  couple  H  for  any 
origin  O  (Fig.  69).  Let  F, 
—  F  be  the  forces,  /  the  arm 
of  the  couple  //",  and  place  this 
couple  so  that  one  of  the  forces, 
say  —  F,  intersects  R  at  O. 
Then,  combining  R  and  —F 
to  their  resultant  F',  the  given 


Fig.  69. 


system  of  forces   is  evidently  equivalent  to  the  two  non-inter- 
secting forces  F,  F'  (compare  Art.  137). 

210.  The  two  forces  F,  F'  determine  "a  tetrahedron  OABC\ 
and  it  can  be  shown  that  the  volume  of  this  tetrahedron  is  con- 
stant and  equal  to  one  sixth  of  the  invariant  of  the  system 
(Art.  207).  The  proof  readily  appears  from  Fig.  69.  The 
volume  of  the  tetrahedron  OABC  is  evidently  one  half  of  the 
volume  of  the  quadrangular  pyramid  whose  vertex  is  C  and 
whose  base  is  the  parallelogram  OBAD.  The  area  of  this 
parallelogram  is  Fp  =  H\  and  the  altitude  of  the  pyramid  is 
=  Rcos(j),  being  equal  to  the  perpendicular  let  fall  from  the 
extremity  of  R  on  the  plane  of  the  couple  ;  hence  the  volume  of 
the  tetrahedron 

=±RH  cos  0= 


211.  To  effect  the  reduction  of  a  given  system  of  forces 
analytically,  it  is  usually  best  to  refer  the  forces  F  and  their 
points  of  application  P  to  a  rectangular  system  of  co-ordinates 


126 


STATICS. 


[211. 


Ox,  Oy,  Oz  (Fig.  70).     Let  x,  y,  z  be  the  co-ordinates  of  P  and 
X,  Y,  Z  the  components  of  F  parallel  to  the  axes. 

To  transfer  these  components  to  O  as  common  origin,  we 
proceed  similarly  as  in  Art.  142.  Thus  to  transfer,  say  X,  we 
introduce  at  P',  the  foot  of  the  perpendicular  let  fall  from  P  on 
the  plane  zx,  two  equal  and  opposite  forces  X,  —  X;  and  we  do 
the  same  thing  at  O.  Then  the  single  force  X at  P  is  replaced 
by  the  force  X  at  O  in  combination  with  the  two  couples  formed 
by  X  at  P,  -X  at  P',  and  X  at  P' ,  -X  at  O.  The  vector  of 
the  former  couple  is  parallel  to  Oz,  its  moment  is  —yX\  the 
negative  sign  being  used  because  for  a  person  looking  on  the 
plane  of  the  couple  from  the  positive  side  of  the  axis  Oz  the 


IP 


-X         P'          X 


Fig.  70. 

couple  rotates  clockwise.  The  vector  of  the  latter  couple  is 
parallel  to  Oy,  and  its  moment  is  zX. 

The  transfer  of  Y  to  the  origin  O  requires  the  introduction  of 
two  couples,  —  zY  having  its  vector  parallel  to  Ox,  and  xY 
having  its  vector  parallel  to  Oz. 

Finally,  transferring  Z  to  O,  we  have  to  introduce  the  couples 
— xZ  witlfla  vector  parallel  to  Oy,  andyZ  with  a  vector  parallel 
to  Ox.  \ 

Thus  each  force  F  is  replaced  by  three  forces  X,  Y,  Z  along 
the  axes  of  io-ordinates  and  applied  at  O,  in  combination  with 
three  couples  whose  vectors  are  yZ—zY  parallel  to  Ox,  zX—xZ 
parallel  to  Oy,  x  Y—yX  parallel  to  Oz. 


2I3-]  CONDITIONS   OF   EQUILIBRIUM.  I2/ 

212.  Doing  the  same  thing  for  every  force  of  the  given  sys- 
tem and  adding  the  components  having  the  same  direction,  the 
system  will  be  found  equivalent  to  the  three  rectangular  forces 


SF, 

applied  at  O,  together  with  the  three  couples 


whose  vectors  are  at  right  angles. 

The   three   forces   can    now   be  compounded   into   a   single 
resultant 


whose  direction  is  determined  by  the  angles  a,  ft,  7.  which  it 
makes  with  the  axes  Ox,  Oy>  Oz, 


^  Y 

Cosa=-—  -,     cos/3=—  ,      cos  7=—- 
K  K  K 

In  the  same  way  the  three  couples  can  be  compounded  into 
a  single  resulting  couple  whose  moment  is 


213.  Since  R*,  as  well  as  //2,  is  thus  found  as  the  sum  of 
three  squares,  each  of  these  quantities  can  vanish  only  if  the 
three  squares  composing  it  vanish  separately.  The  conditions  of 
equilibrium  of  a  rigid  body  (Art.  201)  are  therefore  expressed 
analytically  by  the  following  six  equations  : 


As  the  system  of  co-ordinates  can  be  selected  arbitrarily,  the 
meaning  of  the  first  three  equations  is  that  the  sum  of  the  com- 
ponents of  all  the  forces  along  any  three  lines  not  in  the  same 
plane  must  vanish.  The  last  three  equations  express  that  the 


128  STATICS.  [214. 

sum  of  the  moments  of  all  the  forces  about  any  three  axes  not 
in  the  same  plane  must  also  vanish.  The  moment  of  a  force 
about  an  axis  must  be  understood  as  meaning  the  moment  of  its 
projection  on  a  plane  at  right  angles  to  the  axis  with  respect  to 
the  point  of  intersection  of  the  axis  with  the  plane.  This  defi- 
nition is  in  accordance  with  the  somewhat  vague  notion  of  the 
moment  of  a  force  as  representing  its  "  turning  effect."  For, 
regarding  the  force  as  acting  on  a  rigid  body  with  a  fixed  axis, 
the  force  can  be  resolved  into  two  components,  one  parallel,  the 
other  perpendicular,  to  the  axis  ;  the  former  component  does 
evidently  not  contribute  to  the  turning  effect,  which  is  therefore 
measured  by  the  moment  of  the  latter  alone. 

214.  The  equations  of  the  central  axis  (Art.  204)  can  be  found 
by  a  transformation  of  co-ordinates. 

Let  the  system  be  reduced  for  any  origin  O  to  its  resultant  R, 
whose  rectangular  components  we  denote  by 


and  to  the  vector  //"of  its  resulting  couple  with  the  components 


If  a  point  <9f  whose  co-ordinates  are  f  ,  77,  f  be  taken  as  new 
origin  and  the  co-ordinates  of  any  point  with  respect  to  parallel 
.axes  through  O'  be  denoted  by  x\  y\  z\  we  have  x—^^-x\ 
z-=%+z'.  Substituting  these  values,  we  find 

L  =  2  [  (17  +/)Z-  (?+*')  Y]=^Z-  &  K+2(/Z-*'  Y) 


where  L'  is  the  ^-component  of  the  couple  H*  resulting  for  O' 
as  origin.  Similar  expressions  hold  for  M  and  N.  The  com- 
ponents of  H1  are  therefore 


.and  its  direction  cosines  are 


2IS.] 


CONDITIONS   OF   EQUILIBRIUM. 
p=* 


I29 


L[ 

IJ! 


Hl 


H' 


The  central  axis  being  defined  (Art.  204),  as  that  line  for 
which  the  vector  of  the  resulting  couple  is  parallel  to  the  direc- 
tion of  the  resultant,  the  point  <9'(f,  ??,  £)  will  lie  on  the  central 
axis  if  the  direction  cosines  of  H'  are  proportional  to  those  of 
J?,  viz.,  to 

<*  =  ->    £  =  ->        =-• 

j~)  •  j~)  I  r) 

J\  J\.  I\. 

Hence  the  equations  of  the  central  axis  are 


B  ~    C 


•or 


A 


B 


C 


215.  To  show  the  application  of  the  conditions  of  equilibrium,  let  us 
•consider  the  simple  machine  called  the  wheel  and  axle.  It  consists  of  a 
horizontal  shaft  (Fig.  71)  resting  with  its  ends  on  the  supports  or  bear- 
ings A,  B,  and  is  intended  to  raise  a  weight  W,  suspended  vertically  by 
means  of  a  rope  wound  around  the  shaft.  The  driving  force  F  is  applied 


W 


Fig.  71. 


on  the  circumference  of  the  "wheel,"/.*,  in  a  vertical  plane  at  right 
angles  to  the  axis  of  the  shaft.  It  is  required  to  find  the  relation 
between  F  and  W  for  equilibrium,  and  the  pressures  on  the  bearings 


PART  II — 9 


130  STATICS.  [216. 

Let  r  be  the  radius  of  the  shaft,  R  that  of  the  wheel,  i.e.  the  lever- 
arm  of  the  force  F,  and  let  F  be  inclined  to  the  vertical  at  an  angle  6  ?    , 
then,  with  the  co-ordinates  and  notations  of  the  figure,  the  conditions 
2X=o,  2Y=o,  2Z=o,  give 


A,+Bt=  o,     A,+BV  -  W-Fcos  B  =  o,     Az+£2+Fs'm  (9  =  o, 

where  Ax,  Ay,  A,  are  the  components  of  the  unknown  reaction  at  A  ; 
Bxy  By,  Bs,  those  at  B. 

Taking  moments  about  each  of  the  co-ordinate  axes,  we  find 


FR  =  Wr,     (a  +  6)Fsm  0  +  IBZ=  o,     a  W+  (a+Z>)Fcos  6-lBy=  o, 

where   /=  a  -f-  b  +  c  is  the  length  of  the  shaft. 

Ax  and  Bx  must  evidently  be  separately  zero.     Solving  the  equations,. 
we  find 


216.  As  another  example,  consider  a  rigid  body  of  weight  W,  sup- 
ported at  three  points  Aj,  A2,  A3;  and  let  it  be  required  to  determine  the- 
distribution  of  the  pressure  between  the  three  supports. 

Let  the  vertical  through  the  centroid  of  the  body  meet  the  plane 
of  the  triangle  A^A^A^  in  a  point  G,  whose  distances  from  the  sides 
A2A3,  A3Al}  A^AZ  we  may  denote  by  /lf  /2,  A-  Then,  if  Alf  A2,  Az  be 
the  unknown  reactions,  and  hly  h2,  hz  the  altitudes  of  the  triangle,  we 
have 


and,  taking  moments  about  A2A3,  A3Alr 


Hence,      A,  =      Wy  A,=       W,          A>  =      w. 

M!  "2  "3 

Substituting  these  values  into  the  first  equation,  we  find  the  condition, 


217.] 


CONDITIONS    OF    EQUILIBRIUM. 


If  G  falls  outside  the  triangle,  one  or  two  of  the  points  Aly  A2,  A3 
will  be  subject  to  pressures  vertically  upwards.  If  G  be  the  centroid  of 
the  triangular  area  A^A^,  we  have  p^/h^  —  p^/h^  =/3/^3  =  1/3  ;  hence 
in  this  case  the  three  reactions  are  equal. 

217.  The  axis  of  the  hinges  of  a  door  is  inclined  at  an  angle  6  to  the 
horizon.  The  door  is  turned  out  of  its  position  of  equilibrium  by  an 
angle  <f>,  and  held  in  this  position 
by  a  force  F  perpendicular  to  the 
plane  of  the  door.  Determine  F 
and  the  reaction  of  the  hinges  A, 
B  (Fig.  72). 

Let  the  axis  of  the  hinges  be 
taken  as  the  axis  of  xt  the  verti- 
cal plane  through  it  as  the  plane 
zx,  and  the  point  midway  be- 
tween the  hinges  A,  B  as  the 
origin  O.  Regarding  the  door 
as  a  homogeneous  rectangular 
plate  whose  dimensions  are  AB 
=  2  a,  OC=  2&,  the  co-ordinates 
of  its  centroid  G  are  o,  b  sin  <£, 
b  cos  cf>.  If  the  force  F  be  ap- 
plied at  a  point  Pon  the  middle  line  OC  at  the  distance  OP—p  from 
O,  the  co-ordinates  of  its  point  of  application  P  are  o,  /  sin  <£,  p  cos  <£. 

To  proceed  systematically,  we  may  tabulate  the  components  of  the 
forces,  and  the  co-ordinates  of  their  points  of  application,  and  then 
form  the  component  couples,  as  shown  below.  The  components 
of  the  unknown  reactions  A,  B  of  the  hinges  are  called  Ax,  Ay,  As, 
£*,  B»  Bs. 


Fig.  72. 


FORCES. 

COMPONENTS. 

CO-ORDINATES. 

COUPLES. 

X 

Y 

Z 

X 

y 

z 

yZ-zY 

zX-xZ 

xY-yX 

W 

-  Ws\n  6 

0 

JFcosfl 

o 

b  sin  <£ 

b  cos  (f> 

Wb  cos  9  sin  <j> 

-  Jf2sin0cos<£ 

Wb  sin  6  sin  <J> 

F 

0 

fcos<j> 

—  Fsm<f> 

o 

/>sin<J> 

/cos<|> 

-  Fp(sin*4>  +  cos2<J>) 

o 

o 

A 

n 

Ax 
Bx 

Ay 
By 

A, 
Bz 

a 
—  a 

o 
o 

o 

0 

0 
0 

-A* 

Bga 

Aya 
-Bya 

132  STATICS.  [218. 

From  this  table  the  six  conditions  of  equilibrium  are  at  once  obtained  : 


=  o,  (i) 

=  o,  (2) 

=  o,  (3) 

=o,  (4) 

—  Wb  sin  0  cos  <£  +  (—  Az  +  Bz)a  =  o,  (5) 

=  o.  (6) 


If  the  reactions  were  not  required,  equation  (4)  alone  would  be  suffi- 
cient, as  it  furnishes  the  value  of  F,  viz., 

F=-cos0sm<f>.Wr. 
P 

This  relation  can  of  course  be  found  directly  by  taking  moments  about 
the  axis  of  the  hinges.  It  shows  that,  for  a  given  inclination  of  the  hinges, 
F  is  greatest  when  <£  =  ?r/2. 

The  remaining  five  equations  are  sufficient  to  determine  Ax  +  Bx, 
Ay,  Az,  By,  Bz. 

To  find  the  reactions  for  a  door  with  vertical  axis,  we  have  to  put 
6  =  7T/2,  which  gives,  of  course,  F=o,  and 


Ay-By  =  --  Wsm  4>,     AZ-BZ  =  -- 
a  a 

as  </>  may  be  assumed  =  o  in  this  case,  we  find 


The   signs   indicate   that  the  upper  hinge  A   is  pulled  out  while  the 
lower  one  B  is  pressed  in. 


2.     CONSTRAINTS. 

218.  It  has  been  shown  in  Art.  213  that  the  number  of  the 
conditions  of  equilibrium  is  six,  for  a  rigid  body  that  is  perfectly 
free.  This  number  will  be  diminished  whenever  the  body  is 
subject  to  conditions  restricting  its  possible  motions.  Such 


220.]  CONSTRAINTS.  133 

conditions,  or  constraints,  may  be  of  various  kinds  ;  the  body 
may  have  a  fixed  point,  or  a  fixed  axis,  or  one  of  its  points  may 
be  constrained  to  move  along  a  given  curve  or  to  remain  on  a 
given  surface,  etc. 

As  explained  in  Kinematics,  Art.  37,  a  free  rigid  body  is  said  to 
have  six  degrees  of  freedom.  The  most  general  form  of  motion 
that  it  can  have  is  a  screw-motion,  or  twist,  consisting  of  a  rota- 
tion about  a  certain  axis,  and  a  translation  along  this  axis  ;  each 
of  these  resolves  itself  analytically  into  three  rectangular  com- 
ponents, and  these  six  components  may  be  regarded  as  consti- 
tuting the  six  possible  motions  of  the  body,  on  account  of  which 
it  is  said  to  have  six  degrees  of  freedom. 

Equilibrium  will  exist  only  when  these  six  possible  motions  are 
prevented  ;  hence  there  must  be  six  conditions  of  equilibrium. 

219.  We  proceed  to  consider  some  forms  of  constraint  and 
the  corresponding  changes  in  the  equations  of  equilibrium. 

It  is  generally  convenient  in  dynamics  to  replace  such  restrain- 
ing conditions  by  forces,  usually  called  reactions.  Whenever'it 
is  possible  to  introduce  such  forces  having  the  same  effect  as 
the  given  conditions,  the  body  may  be  regarded  as  free,  and  the 
general  equations  of  equilibrium  can  be  applied. 

Before  considering  the  constraints  of  a  rigid  body,  those  of  a 
single  particle,  or  point,  must  be  briefly  discussed. 

220.  Particle  constrained  to  a  Surface.     A   free   particle  has 
three  degrees  of  freedom  ;   and  accordingly  its   equilibrium   is 
determined  by  three  conditions  (Art.  101)  : 

o.  (i) 


If  the  co-ordinates  determining  the  position  of  the  particle  be 
subject  to  one  condition,  expressed  by  an  equation  between  these 
co-ordinates,  the  particle  is  said  to  have  two  degrees  of  freedom 
and  one  constraint.  Its  motion  is  restricted  to  the  surface  repre- 
sented by  the  equation  between  its  co-ordinates,  say 

4>(x,y,z)=o.  (2) 


I34  STATICS.  [221. 

The  condition  that  the  particle  should  remain  on  this  surface 
can  be  replaced  by  introducing  the  reaction  of  the  sujface,  i.e.  a 
force  that  is  always  so  directed  as  not  to  allow  the  particle  to 
leave  the  surface.  Combining  this  force  with  the  given  forces 
acting  on  the  particle,  this  particle  can  be  regarded  as  free,  and 
the  general  conditions  of  equilibrium  must  hold. 

221.  If  the  surface  be  smooth,  i.e.  if  the  particle  move  along 
it  without  friction,  the  reaction  of  the  surface  must  be  directed 
along  the  normal  to  the  surface  (2).  Let  N  denote  this  normal 
reaction  ;  Nx,  Nv,  Ng  its  components  ;  then  the  conditions  of 
equilibrium  are 

2X+N,  =  o,     2F+^  =  o,     2Z+^.  =  o.  (3) 

The  condition  that  N  has  the  direction  of  the  normal  is 
expressed  by  the  relations 


where  <£,,=  -,     <£y=-2,    <^=_     are  obtained  from  (2). 
ox  oj/  dz 

Eliminating  the  reactions  by  means  of  (3),  we  find  the  two  con- 
ditions of  equilibrium, 


<#>»  </>y  </>* 

The  meaning  of  these  equations  is  obvious  ;  they  express  that 
the  resultant  of  the  given  forces  must  have  the  direction  of  the 
normal  to  the  surface. 

The  problem  generally  consists  in  finding  the  positions  of 
equilibrium  of  the  particle  on  the  surface.  The  two  equations 
(5)  represent  a  curve  whose  intersections  with  the  surface  (2) 
give  the  required  positions. 


222.]  CONSTRAINTS. 

The  magnitude  of  the  reaction  N  is  found  from  (3)  : 


222.  If  the  surface  be  rough,  the  total  reaction  of  the  surface 
lies  within  the  cone  of  friction  (Art.  187),  and  the  resultant  R  of 
all  other  forces  acting  on  the  particle  must  therefore  also  fall 
within  this  cone. 

The  boundaries  of  the  regions  on  the  surface  within  which 
equilibrium  is  possible  are  .found  by  considering  the  total  reac- 
tion in  its  limiting  position,  i.e.  when  it  makes  the  friction  angle 
tan"1//,  with  the  normal  to  the  surface. 

Let  N  represent  the  normal  component  of  the  total  reaction  ; 
Nx)  Ny,  Ng  its  components  ;  the  force  of  friction  pN  lies  in  the 
tangent  plane,  and  has,  therefore,  the  components  pNdx/ds, 
l*,Ndy/ds,  pNdz/ds,  for  motion  along  any  curve  s  on  the  sur- 
face (2).  These  components  of  the  friction  must  be  given  the 
double  sign  T,  because  the  force  of  friction  may  act  in  either 
sense  along  the  curve  s.  Thus,  the  conditions  of  equilibrium 
are 

—  =o,, 
as 


o,  (6) 

as 


. 
as 

To  eliminate  the  reactions,  multiply  these  equations  by  <f>x, 
<f>,  and  add  ;  this  gives 


since  the  differentiation  of  the  equation  of  the  surface  (2)  gives 
$xdx+$ydy  +  $zdz=v.  Substituting  for  Nx,  Nv,  N,  their  values 
from  (4),  the  equation  becomes 


(7) 


136  STATICS.  [222. 

This  equation  determines  the  normal  reaction  N  of  the 
surface. 

To  obtain  an  expression  .for  pN,  multiply  the  second  of  the 
equations  (6)  by  (f>z,  the  third  by  <f>y  and  subtract  ;  owing  to 
the  relations  (4)  this  gives 


Similarly,  we  find 


-2  K-  $  =  ± 

The  left-hand  members  as  well  as  the  parentheses  on  the  right 
are  determinants  of  the  second  order;  hence,  squaring  and 
adding,  we  find 


(8) 


If  TVbe  now  eliminated  between  (7)  and  (8),  we  find  the  final 
condition  of  equilibrium  that  must  be  fulfilled  by  the  given  forces 
independently  of  the  reaction  of  the  surface  : 


Putting  this  equation  into  the  form 
i 


R     R'       R     R'      R 


where  ^2=(2^)2-f(2F)2+(2Z)2  and  ^'2  =  0x2  +  0y2  +  ^2,  it  is 
seen  to  express  the  fact  that  the  resultant  R  of  the  given  forces 
makes  the  friction  angle  0  with  the  normal,  each  member  of  the 
equation  being  an  expression  for  the  cosine  of  this  angle. 


224-] 


CONSTRAINTS. 


137 


The  regions  of  the  surface  (2)  on  which  the  particle  is  in 
equilibrium  are  cut  out  of  this  surface  by  the  surface  (9). 

223.   Particle  constrained  to  a  Curve.     If  the  particle  be  sub- 
ject to  two  conditions, 


$  (x,  y,  *)  =o,     i/r  (x,  7,  z)  =  0, 


(10) 


so  that  it  has  two  constraints  and  but  one  degree  of  freedom,  its 
motion  is  restricted  to  the  curve  of  intersection  of  the  two  sur- 
faces (10).  The  particle  may  be  imagined  as  a  small  sphere 
moving  within  a  tube,  or  as  a  small  ring  or  bead  sliding  along  a 
thin  wire. 

224.  Let  .the  curve  be  smooth,  so  that  its  total  reaction  is 
along  the  normal  to  the  curve.  Denoting  this  normal  reaction 
again  by  N,  its  components  by  Nt1  Ny,  JVZ)  the  conditions  of 
equilibrium  are 

:0)  (II) 


or 


(12) 


The  condition  that  N  has  the  direction  of  the  normal  can  be 
expressed  in  the  form 

Nxdx  +  Nydy  +  Ntdz  =  o, 
which,  by  (12),  reduces  to  fc 


Differentiating  the  equations  of  the  curve  (10),  we  find 


and  eliminating  the  differentials  between  the  last  three  equa- 
tions, the  single  condition  of  equilibrium,  independent  of  the 
reactions,  is  found  in  the  form 


STATICS. 


[225. 


(13) 


The  intersections  of  the  surface  (13)  with  the  curve  (10)  give 
the  positions  of  equilibrium  of  the  particle  on  the  curve. 

The  reaction  of  the  curve,  or  the  pressure  on  the  curve  which 
is  equal  and  opposite  to  this  reaction,  can  then  be  found  from 
the  equations  (11). 

225.  For  a  rough  curve,  the  total  reaction  resolves  itself  into 
.a  normal  component  N  and  a  tangential  component  pN,  which 
represents  the  frictional  resistance.  The  equations  of  equi- 
librium are 


(14) 


ds 


Transposing  the  third  terms,  multiplying  by  dx/ds,  dy/ds, 
)  and  adding,  we  find,  since  Nxdx+Nydy-\-Ngdz  —  v, 


as 


as 


as 


(is) 


Multiplying  the  second  of  the  equations  (14)  by  dzjds,  the 
third  by  dy/ds,  and  subtracting,  we  have 


ds 


ds 


ds 


•    -11  ^  ^    dx     ^  v  dz         (  \r  dx      ,r  dz\ 

similarly,         zZ  -  —  —  2,X-  —  =  —(Nz-  --  Nx—  - 
ds  ds          \     ds  ds) 


ds 


ds 


,- 

ds          ds 


226.]  CONSTRAINTS. 

Each  member  being  a  determinant  of  the  second  order,  we  find 
by  squaring  and  adding  the  three  equations, 


ds         .as  ds 

The  reaction  N  can  now  be  eliminated  between  (15)  and  (16), 
and  we  obtain  the  single  condition  of  equilibrium  independent 
of  the  reaction  : 


_  _ 
ds  ds  ds          Vi  -fyu,2 

(17) 

The  differential  coefficients  dx/ds,  dy/ds,  dz/ds  must  satisfy 
the  differential  equations  of  the  curve  (10),  viz.  : 

dx  .    ,  dy  .    ,   dz 

d-s+*'i+*'^=0> 

dx  .    ,   dy  .    ,   dz 
- 


If  the  values  of  dx/ds,  dy/ds,  dz/ds  be  determined  from  the 
last  three  equations  and  substituted  into  the  relation 


the  equation  of  a  surface  will  result,  which  cuts  out,  on  the  curve 
(10),  the  limits  between  which  equilibrium  is  possible. 

226.  Rigid  Body  with  a  Fixed  Point.  A  body  that  is  free  to 
turn  about  a  fixed  point  A  can  be  regarded  as  free  if  the  reaction 
A  of  this  point  be  introduced  and  combined  with  the  other 
forces  acting  on  the  body. 

Let  Ax,  AyJ  Ag  be  the  components  of  A  ;  then,  taking  the  fixed 
point  A  as  origin,  the  six  equations  of  equilibrium  (Art.  213)  are 


I40  STATICS.  [227. 

The  first  three  of  these  equations  serve  to  determine  the 
reaction  of  the  fixed  point  ;  the  last  three  are  the  actual  con- 
ditions of  equilibrium  corresponding  to  the  three  degrees  of 
freedom  of  a  body  with  a  fixed  point. 

Hence,  a  rigid  body  having  a  fixed  point  is  in  equilibrium  if 
the  sum  of  the  moments  of  all  the  forces  vanishes  for  any  three 
axes  passing  through  the  fixed  point  and  not  -situated  in  the  same 
plane. 

227.  Rigid  Body  with  a  Fixed  Axis,  A  body  with  a  fixed  axis 
has  but  one  degree  of  freedom  ;  indeed,  the  only  possible  motion 
consists  in  rotation  about  this  axis. 

An  axis  is  fixed  as  soon  as  two  of  its  points,  say  A,  B,  are 
fixed.  Hence,  introducing  the  reactions  At,  Ay,  Agt  B#  By,  Bz 
of  these  points,  the  body  can  be  regarded  as  free.  If  the  point  B 
be  taken  as  origin,  the  line  BA  as  axis  of  z  (Fig.  73),  the  equa- 
tions of  equilibrium  become 


B 


IA, 


where  a  =  BA. 

The  last  of  the  six  equations  is  the  only  independent  condi- 
tion of  equilibrium  of  the  con- 
strained body ;  the  first  five 
determine  Ax,  B&  Ay,  By,  Ax 
+  Bg.  The  two  ^-components 

_       , ; cannot    be    found    separately, 

/ "R "7 A 
jf  '      jf  since   they   act    in   the   same 

^/D«  'Ay 

/  straight  line. 

Fig.  73. 

Hence,  a  rigid  body  having- 

a  fixed  axis  is  in  equilibrium  if  the  sum  of  the  moments  of  all  the 
forces  vanishes  for  the  fixed  axis. 

228.  If,  in  the  preceding  article,  the  axis  be  not  absolutely 
fixed,  but  only  fixed  in  direction  so  that  the  body  can  rotate  about 
the  axis  and  also  slide  along  it,  we  have  evidently 


229.]  CONSTRAINTS. 


hence,  by  the  third  equation  of  equilibrium, 


as  an  additional  condition  of  equilibrium. 

The  body  has  in  this  case  two  degrees  of  freedom. 

229.  Rigid  Body  with  a  Fixed  Plane.  A  body  constrained  to 
slide  on  a  fixed  plane  has  three  degrees  of  freedom.  At  every 
point  of  contact  between  the  body  and  the  plane,  the  latter 
exerts  a  reaction.  As  all  these  reactions  are  parallel,  they  can 
be  combined  into  a  single  resultant  N.  Taking  the  fixed  plane 
as  the  plane  xy,  TV  will  be  parallel  to  the  axis  of  z\  hence,  if  a, 
b,  o  be  the  co-ordinates  of  its  point  of  application,  the  six 
equations  of  equilibrium  are 


The  third,  fourth,  and  fifth  equations  determine  the  reaction 
N  and  the  co-ordinates  a,  b  of  its  point  of  application.  The 
three  other  equations  are  the  actual  conditions  of  equilibrium  ; 
they  agree,  of  course,  with  the  three  conditions  of  equilibrium 
of  a  plane  system  as  found  in  Art.  143. 

If  there  be  not  more  than  three  points  of  contact  (or  supports) 
between  the  body  and  the  fixed  plane,  the  reactions  of  these 
points  can  be  found  separately.  Let  Av  A2,  A3  be  the  three 
points  of  contact  ;  Nv  N^  N3  the  required  reactions  ;  av  b^ 
#2,  bv  as,  £3  the  co-ordinates  of  Av  A2,  A3  ;  then  N  must  be 
resolved  into  three  parallel  forces  passing  through  these  points, 
and  the  conditions  are 


a^  4-  0 


142 


STATICS. 


[230. 


These  three  equations  determine  N^  N^  A^,  unless  the  three 
points  AV  A2,  A3  be  situated  in  a  straight  line  ;  for  in  this  case 
the  determinant  of  the  coefficients  of  Nlt  N^  Nz  vanishes, 


i       I 


bl 


=o. 


The  reactions  become  almost  indeterminate  whenever  there 
are  more  than  three  points  of  contact. 

230.  In  addition  to  the  works  of  ROUTH,  MINCHIN,  PRICE,  TODHUNTER 
mentioned  in  Art.  199,  the  student  is  referred,  in  particular  for  the 
more  advanced  parts  of  the  subject,  to  W.  SCHELL,  Theorie  der  Bewe 
gung  und  der  Krafte,  Vol.  II.,  Leipzig,  Teubner,  1880;  A.  F.  MOBIUS 
Lehrbuch  derStatik,  Leipzig,  Goschen,  183 7,  reprinted  in  MOBIUS'S  Gesam 
melte  Werke,  Vol.  III.,  Leipzig,  Hirzel,  1886  ;  MOIGNO,  Statique,  Paris 
Gauthier-Villars,  1868  ;  J.  SOMOFF,  Theoretische  Mechanik,  iibersetzt  von 
A.  Ziwet,  Vol.  II.,  Leipzig,  Teubner,  1879  ;  E.  COLLIGNON,  Statique 
Paris,  Hachette,  1889  ;  THOMSON  AND  TAIT,  Natural  Philosophy,  Part.  II. 
Cambridge,  University  Press,  1890. 


232.]  THE   PRINCIPLE   OF   VIRTUAL  WORK.  143, 


VII.    The  Principle  of  Virtual  Work. 

231.  Work  has  been  defined  in  Art.  72  as  the  product  of  a 
force  into  the  displacement  of  its  point  of  application  in  the 
direction  of  the  force. 

Thus  the  expansive  force  F  of  the  steam  in  the  cylinder  of  a 
steam-engine,  in  pushing  the  piston  through  a  distance  s,  is  -said 
to  do  work,  and  this  work  is  measured  by  the  product  Fs.  Simi- 
larly the  force  of  gravity,  i.e.  the  attractive  force  of  the  earth's 
mass,  does  work  on  a  falling  body. 

The  resistance  to  be  overcome  by  the  engine,  in  the  former 
case,  and  the  resistance  of  the  air  in  the  latter,  are  also  forces 
acting  on  the  body  during  its  displacement.  But  as  the  sense 
of  the  displacement  is  opposite  to  that  of  these  forces,  their 
work  is  negative ;  work  is  done  against  these  forces.  Thus  the 
muscular  force  of  a  man  who  raises  a  weight  does  work  against 
gravity ;  if  the  weight  he  holds  is  so  heavy  as  to  pull  him  down, 
gravity  does  work  against  his  force  ;  if  he  merely  tugs  at  a 
weight  without  being  able  to  lift  it,  the  work  is  zero,  because 
the  displacement  is  zero. 

232.  In  general,  the  point  of  application  of  a  force  /''will  be 
acted  upon  by  a  number  of  different  forces,  so  that  the  displace- 
ment s  of  this  point  will  not  necessarily  take  place  in  the  direc- 
tion of  F.     In  this  general  case  the  work  of  a  force  is  defined  as 
the  product  of  the  force  into  the  projection  of  the  displacement  of 
its  point  of  application  on  the  direction  of  the  force. 

In  Fig.  74,  for  instance,  the  particle  P  while  acted  upon  by 
the  force  F  (and  any  number  of  other 
forces)  is  displaced  from  P  to  P1 ;  hence 
if  PP'  =  s,  and   KP'PQ  =  $,  the  work 
of  the  force 


(I) 

It  is  obvious  that  this  work  might  also  be  defined  as  the 


144  STATICS.  [233, 

product  of  the  displacement  into  the  projection  of  the  force  on 
the  displacement ;  for  we  have 

Fscos<t>  =  F-PQ  =  s>PR. 

The  work  of  a  force  is  evidently  positive  or  negative  accord- 
ing as  the  angle  </>  is  less  or  greater  than  Tr/2,  provided  we 
select  for  <£  always  that  angle  between  F  and  s  which  is  not 
greater  than  TT. 

233.  The  above  definition  of  work  assumes  that  the  force  F 
remains  constant,  both  in  magnitude  and  direction,  while  the  dis- 
placement s  takes  place,  and  that  this  displacement  is  recti- 
linear. If  either,  or  both,  of  these  conditions  be  not  fulfilled, 
the  definition  can  be  applied  only  to  infinitesimal  displacements 
ds.  As  the  work  done  by  a  finite  force  F  during  such  a  dis- 
placement ds  is  infinitesimal,  we  have 

and  the  total  work  done  by  any  variable  force  F  while  its  point 
of  application  is  displaced  along  any  straight  or  curvilinear  patl 
?,  is  obtained  by  integrating  from  P  to  Q  : 


234.    Since  work  can  always  be  regarded  as  the  product  of 
force  into  a  length,   its  dimensions  are  found  by  multiplying 
those  of  force,  MLT~2  (Art.  64),  by  L  ;  hence,  the  dimensions  oj 
work  are 


The  unit  of  work  is  the  work  of  a  unit  force  (poundal,  dyne) 
through  a  unit  distance  (foot,  centimetre).     The  unit  of  work  ii 
the  F.P.S.  system  is  called  the  f  oot-poundal  ;  in  the  C.G.S.  sy 
tern,  the  erg.     Thus,  the  erg  is  the  amount  of  work  done  by 
force  of  one  dyne  acting  through  a  distance  of  one  centimetre. 
These  are  trie  absolute  units. 


236.]  THE   PRINCIPLE   OF   VIRTUAL  WORK.  145 

In  the  gravitation  system  where  the  pound,  or  the  kilo- 
gramme, is  taken  as  unit  of  force,  the  British  unit  of  work  is 
the  foot-pound,  while  in  the  metric  system  it  is  customary  to 
use  the  kilogramme-metre  as  unit. 

235.  The  numerical  relations  between  these  units  are  obtained 
as  follows.  Let  x  be  the  number  of  ergs  in  the  foot-poundal, 
then  (comp.  Art.  66), 

gm.  cm.2  _      Ib.  ft.2 
x  •  -  -     —  I  •  -  —  > 
sec.2  sec.2 

hence          *=^'(^)  =453'59><  3O-47972=4-2i39X  io5  ; 

i.e.  i  foot-poundal  =  4.  2139  x  i  o5  ergs,  and  I  erg=2.3/2i  x  IO"6 
=  0.000002  372  i  foot-poundal. 

Again,  let  x  be  the  number  of  kilogramme-metres  in  I  foot- 
pound, then 

.  m.  =  i  ft.  Ib., 


hence          ^=-^-—=0.45359x0.3048=0.13825, 
kg.   m. 

i.e.  i  foot-pound  =0.13825  kilogramme-metres. 

Finally,  i  foot-pound  =g  foot-poundals  (Art.  69)  ;  hence  i  foot- 
pound =  1.3  56  x  io7  ergs,  and  i  erg  =  7.  3737  x  io~8  foot-pounds, 
if  ^=32.2. 

236.   Exercises. 

(1)  A  joule  being  defined  as  io7  ergs,  show  that  i  foot-pound  =  1.356 
joules,  and  that  i  joule  is  about  3/4  foot-pound. 

(2)  Show  that  a  kilogramme-metre  is  nearly  io8  ergs. 

(3)  What  is  the  work  done  against  gravity  in  raising  300  Ibs.  through 
a  height  of  25  ft.  :   (a)  in  foot-pounds,  (b)  in  ergs? 

(4)  Find  the  work  done  against  friction  in  moving  a  car  weighing  3 
tons  through  a  distance  of  fifty  yards  on  a  level  road,  the  coefficient  of 
friction  being  0.02. 

PART    II  —  10 


I46  STATICS.  [237. 

(5)  A  mass  of  12  Ibs.  slides  down  a  smooth  plane  inclined  at  an  angle 
of  30°  to  the  horizon,  through  a  distance  of  25  ft.  ;  what  is  the  work 
done  by  gravity  ? 

237.  It  follows  from  the  definition  of  work  that,  if  any  num- 
ber of  forces  Fv  F2,  .  .  .,  Fn  act  on  a  particle  P,  the  sum  of  their 
works  for  any  displacement  PP'  =  ds  is  equal  to  the  work  of  their 
resultant  R  for  the  same  displacement.  For,  the  resultant  R 
being  the  closing  line  of  the  polygon  constructed  by  adding  the 
forces  Fv  F2,  ...,  Fn  geometrically,  the  projection  of  R  on  any 
direction,  such  as  PP',  is  equal  to  the  sum  of  the  projections  of 
the  forces  F  on  the  same  line  (Art.  89)  ;  that  is,  if  <*15  «2,  .  .  ., 
«n  be  the  angles  made  by  Fv  F2,  .  .  .,  Fn  with  PP',  and  a  the 
angle  between  R  and  PP',  we  have 

F1  cos  «!  +  F2  cos  «2  H  -----  \-Fn  cos  «n  =  R  cos  a  ; 

multiplying  this  equation  by  ds,  we  obtain  the  above  proposition 

F1  cos  oL^ds  +  F2  cos  v^ds  -\  -----  \-  Fn  cos  ands  =  R  cos  ads, 


which   expresses   the   so-called   principle  of  work  for  a  single 
particle. 

238.  When  the  particle  is  in  equilibrium,  so  that  the  forces 
do  not  actually  change  the  motion,  we  may  derive  from  this 
proposition  a  convenient  expression  for  the  conditions  of  equi- 
librium by  considering  displacements  that  might  be  given  to 
the  particle.     Such  displacements  are  called  virtual,  and  the 
corresponding  work  of  any  of  the  forces  is  called  virtual  work. 

It  is  customary  to  denote  a  virtual  displacement  by  Ss,  the 
letter  B  being  used  to  distinguish  from  an  actual  displacement 
ds  ;  this  distinction  becomes  of  importance  in  kinetics. 

239.  The  resultant  being  zero  in  the  case  of  equilibrium,  the 
sum  of  the  virtual  works  of  all  forces  acting  on  the  particle  must 
be  zero  for  any  virtual  displacement,  i.e. 


^i  cos  «!&$•  -f  Fz  cos  aJBs  -\  -----  h  FH  cos  an$s  =  o.  (4)  , 

As  the  resultant  must  vanish  if    its  three  projections    vanish 
for  any  three  axes  not  lying  in  the  same  plane,  the  necessary 


240.]  THE   PRINCIPLE   OF  VIRTUAL  WORK.  147 

and  sufficient  conditions  of  equilibrium  of  a  single  particle  are 
that  the  sum  of  the  virtual  zvorks  of  all  forces  must  be  zero  for 
any  three  virtual  displacements  not  all  in  tJie  same  plane. 
This  is  the  principle  of  virtual  work  for  a  single  particle. 

If  the  particle  be  referred  to  a  rectangular  system  of  co-ordi- 
nates, its  displacement  8s  can  be  resolved  into  three  com- 
ponent displacements  &r,  by,  Sz,  parallel  to  the  axes.  The 
forces  acting  on  the  particle  being  replaced  by  their  compo- 
nents X,  Y,  Z,  the  sum  of  their  virtual  works,  for  the  displace- 
ment Bs  is  2-Y-&M-2K-5y  +  2Z.&sr.  Hence  the  analytical 
expression  for  the  principle  of  virtual  work  : 

$X-  &r+2  Y-  Sy  +  2Z-Sz  =  0.  (5) 

As  the  displacements  &r,  fy,  8z  are  independent  of  each  other, 
and  perfectly  arbitrary,  this  single  equation  is  equivalent  to  the 

three  equations 

o,     2F=o, 


which  are  the  ordinary  conditions  of  equilibrium  of  a  single 
particle. 

240.  The  principle  of  virtual  work  is  particularly  useful  in 
eliminating  the  unknown  reactions  arising  from  constraints. 

Suppose  the  particle  be  constrained  to  a  smooth  surface  or 
curve.  After  introducing  the  normal  reaction  of  the  surface 
or  curve  the  particle  can  be  regarded  as  free  ;  and  the  equation 
of  virtual  work  can  be  used  to  express  the  conditions  of  equi- 
librium. This  equation  will,  in  general,  contain  the  unknown 
reaction.  But  as  this  reaction  has  the  direction  of  the  normal, 
it  will  be  eliminated  if  the  virtual  displacement  be  selected 
along  a  tangent.  Hence,  immhe  case  of  constrainment  to  a  sur- 
face, the  two  conditions  of  equilibrium  independent  of  the  reaction 
are  found  by  forming  the^^uiation  of  virtual  work  for  virtual 
displacements  along  any  tw^^mLgents  to  the  surface  ;  and  in  the 
case  of  constrainment  to  a  40^  the  one  suck  condition  is  found 
from  a  virtual  displacement  along  the  tangent. 

If  it  be  required  to  find  the  normal  pressure  on  the  surface 


148 


STATICS. 


or  curve,  which  is  of  course  equal  and  opposite  to  the  reaction, 
it  can  be  found  from  a  virtual  displacement  along  the  normal. 

241.  If  the  equation  (4)  which  expresses  the  principle  of  vir- 
tual work  be  divided  by  the  element  of  time  &*,  during  which 
the  displacement  8s  would  take  place,  the  factor  §s/$t=v  repre- 
sents a  virtual  velocity,  and  the  equation  becomes 

F1  cos  «!  •  v  -f  F2  cos  «2  •  v  -\ h  Fn  cos  an  •  v  =  o. 

On  account  of  this  form,  the  proposition  is  often  called  the 
principle  of  virtual  velocities. 

The  product  of  a  force  into  the  virtual  velocity  of  its  point  of 
application  in  the  direction  of  the  force,  Fcosct'V,  is  some- 
times called  the  virtual  moment  of  the  force. 

242.  The  principle  of  virtual  work  can  readily  be  extended  to 
the  case  of  a  rigid  body  acted  upon  by  any  number  of  forces. 

The  forces  acting  on  a  rigid  body  can  always  be  reduced  to  a 
resultant  R  and  a  resulting  couple  H  (Art.  200).  This  reduc- 
tion is  based  on  the  supposition  (Art.  84)  that  the  point  of 
application  of  a  force  can  be  displaced  arbitrarily  along  the  line 

of  the  force.  It  can  be  shown  that 
such  a  displacement  of  the  point  of  ap- 
plication P  of  a  force  F  (Fig.  75),  from 
P  to  Q  along  the  line  of  the  force,  does 
not  affect  the  work  done  by  the  force 
in  any  infinitesimal  displacement  of  the 
body.  Let  PF'  =  Ss  be  the  displace- 
ment of  j?,  QQ'  =  Ss'  that  of  g;  let  p 
and  q  be  the  projections  of  P  and  Q  on 
the  line  (•the  force  F\  then,  since  the 
body  is  rigid,  P'Q'  =  PQ\  and  conse- 
quently Q^Kill ^differ  from  Pp  only  by 
an  infinitesimal  of  an  order  hi^HF  than  the  order  of  the  dis- 
placement PP'  =  8s.  Hence,  ^p 

F-Pp=F.  Qq. 


Fig.  75. 


243-]  THE   PRINCIPLE   OF   VIRTUAL  WORK.  149 

It  may  here  be  noted  that,  in  general,  the  principle  of  virtual 
work  must  be  understood  to  mean  that  the  sum  of  the  works 
of  the  forces  differs  from  the  work  of  their  resultant  by  an 
infinitesimal  of  an  order  higher  than  that  of  the  virtual  dis- 
placement. It  does  not  mean  that  the  difference  is  absolutely 
zero. 

243.  Owing  to  the  proposition  proved  in  the  preceding  article, 
the  sum  of  the  works  of  all  the  forces  acting  on  a  rigid  body  is 
equal  to  the  sum  of  the  works  of  the  resultant  R  and  the  result- 
ing couple  H  for  any  infinitesimal  displacement  of  the  body, 
and  the  work  of  the  forces  is  not  changed  by  such  a  displace- 
ment. 

It  follows  that  the  necessary  and  sufficient  conditions  of 
equilibrium  of  a  rigid  body  (Art.  201),  viz. 

^  =  0,  H=o, 

can  be  expressed  by  saying  that  the  sum  of  the  virtual  works 
of  all  the  forces  must  be  zero  for  any  infinitesimal  displacement 
of  the  body. 

For  when  the  forces  are  in  equilibrium,  this  condition  is 
evidently  fulfilled.  To  prove  that  there  must  be  equilibrium 
whenever  this  condition  is  fulfilled,  it  is  only  necessary  to  show 
that  both  R  and  H  must  vanish  if  the  sum  of  their  works  is 
zero  for  any  infinitesimal  displacement.  A 

To  see  this,  consider  first  a  displacement  of  translation,  Ss, 
parallel  to  R.  The  work  of  R  will  be  RSs  while  the  works  of 
the  two  forces  constituting  H  are  equal  and  opposite,  so  that 
the  work  of  H  is  zero.  As  the  sum  of  the  works  of  R  and  H 
must  vanish  by  hypothesis,  it  follows  that  R  =  o. 

Next  consider  a  displacement  of  rotation  S0  about  an  axis 
parallel  to  the  vector  H.  Taking  this  axis  so  as  to  intersect 
R  and  bisect  the  arm/  of  ttffcouple  //",  the  work  of  R  will  be 
zero  while  that  of  each  of  «  forces  F  of  the  couple  H  will 
hence  the  whole  work  of  H  is  FW  =  HW.  As 


150  STATICS.  [244. 

the  sum  of  the  works  of  R  and  H  must  vanish  by  hypothesis, 
it  follows  that  H=o. 

The  two  conditions  R  =  o,  H=o,  are,  therefore,  both  fulfilled. 

244.  The  following  examples  may  serve  to  illustrate  the  appli- 
cation of  the  principle  of  virtual  work. 

To  find  the  force  just  necessary  to  move  a  cylinder  of  radius  r  and 
weight  W  up  a  plane  inclined  at  an  angle  a  to  the  horizon  by  means  of  a 
crow-bar  of  length  \  set  at  an  angle  (3  to  the  horizon  (Fig.  76). 

Let  s  be  the  distance  from  the  fulcrum  A  of  the  crow-bar  to  the 

point  of  contact  B  of  the  cylinder 
with  the  plane. 

Turning  the  crow-bar  about  A  by 
an  angle  8ft,  the  work  of  the  force  F 
acting  at  the  end  of  the  bar  is  F-  1  8ft. 
The  corresponding  displacement  of 
the  centre  C  of  the  circle,  which  is 
the  point  of  application  of  the  force 
W,  is  parallel  to  the  inclined  plane, 
and  may  be  regarded  as  the  differen- 
tial 8s  of  the  distance  AB  =  s.  The 
work  of  W  is,  therefore,  W8s  sin  a.  This  gives  the  equation  of  work 


hence 

"i—  i- 

The  relation  between  s  and  ft  can  be  found  by  projecting  ABCDA 
on  the  vertical  line  ;  this  gives 

r  cos  a  -f-  s  sin  a  =  r  cos  ft  +  J  sin/3, 

whence  ,  =  rco»0-cos«. 

sin  a  —  sin  ft 

Differentiating  the  former  equation,  we  find 

sin  a8s  =  -  r  sin  (38ft  +  s  cos  ft  8ft  +  sin  ft  8s, 


.    8s  __  scosft  —  r  sin  ft  _  ^cos2/?  —  cos«  cos  ft  —  sin  a  sin/?  +  sin2/? 
8ft         since  —  sin/3  (since  —  sin  ft)'2 


THE   PRINCIPLE   OF    VIRTUAL  WORK. 


>or 


I     8s  ___  I  —  COS(«  —  /?)  I 

r  8J3  ~  (sin  a-  sin/2)2  =  i  +cos(a 


Hence,  finally, 


245.  A  weightless  rod  of  length  AB  =  1  rests  at  C  on  a  horizontal 
.cylinder  whose  axis  is  at  right  angles  to  the  vertical  plane  through  the  rod ; 
its  lower  end  A  leans  against  a  vertical  wall,  and  from  its  upper  end  B  a 
weight  W  is  suspended.  Determine  the  reactions  at  A  and  C,  and  the 
distance  AC  =  *for  equilibrium,  if  the  distance  CD  =  a  of  the  point  of 
.support from  the  vertical  wall  is  given  (Fig.  75). 

(a)  Let  A  glide  vertically  upwards,  C  remaining  in  contact.  At  A 
.as  well  as  at  C  the  forces  are  perpendicular  to  the  displacements; 
Jience,  putting  EB  =y,  we  have  Wfy  =  o. 

,C 


whence,  (/—  .x).*2  —  /(JT  —  #2)  =  o 


Fig.   77. 


•or 


(b)  Give  the  rod  a  vertical  displacement  to  a  parallel  position : 


(c)  Give  the  rod  a  displacement  in  its  own  directign  : 


246.  /«  ^  parallelogram  formed  by  four  rods  with  hinges  at  the  ver- 
Jices,  elastic  strings  are  stretched  along  the  diagonals.  Determine  the 
ratio  of  the  tensions  in  these  strings* 


152  STATICS.  [247. 

Let  m,  m1  be  the  lengths  of  the  diagonals,  T,  T'  the  tensions,  and 
$m,  8m'  the  changes  of  length  of  the  diagonals  when  the  parallelogram 
is  slightly  deformed  ;  then  by  the  principle  of  virtual  work 


o.  (i) 

From  geometry  we  have,  if  a,  b  are  the  sides  of  the  parallelogram, 


hence,  differentiating,         m  8m  +  m'  8m'  =  o.  (2) 

From  (i)  and  (2)  we  find 

7/7"  =  */*'.  (3) 

247.  For  the  purposes  of  statics  a  body  is  regarded  as  rigid 
if  the  points  of  application  of  all  the  forces  acting  on  the  body 
have  invariable  distances  from  each  other;  these  points  may 
be  imagined  connected  by  a  framework  of  rigid  rods.  The 
tensions  in  these  connecting  rods,  since  they  occur  in  pairs  of 
equal  and  opposite  forces  acting  along  rigid  lines,  do  not  enter 
into  the  equation  of  virtual  work.  One  of  the  chief  advantages 
of  the  principle  of  virtual  work  consists  in  this  elimination  of 
these  internal  forces. 

Let  us  now  generalize  the  idea  of  the  rigid  body  by  assuming^ 
the  points  of  application  of  the  forces  to  be  connected  by  rods- 
or  threads  which  may  even  be  elastic  ;  the  points  may  also  be 
constrained  to  move  on  smooth  surfaces  or  curves  ;  friction, 
however,  is  to  be  excluded. 

Let  x,  y,  z  be  the  co-ordinates  of  one  of  the  points,  P  \ 
x'  t  _/,  z't  those  of  another  point,  Q  ;  let  /  be  the  length  of  the 
connecting  thread  or  rod,  PQ  ;  «,  ft,  7,  its  direction  cosines  ; 
and  let  T  be  the  tension  or  stress  in  PQ.  If  the  whole  system 
be  subjected  to  any  infinitesimal  displacement,  for  which  &r,  fry, 
Sz  are  the  component  displacements  of  P,  8x',  S/,  Bzr  those  of 
Q,  the  sum  of  the  works  of  the  two  equal  and  opposite  forces  T 
for  this  displacement  will  be 

To. 


248.]  THE   PRINCIPLE   OF   VIRTUAL  WORK. 

or,  since  a  =  (x-x')/l,  /3  =  (/-/)//,  -f=(z-z')/l, 


Differentiating  the  relation 


we  have 


hence  the  sum  of   the  virtual  works  of  the  two  tensions   T 
reduces  to 

rs/; 

which  is  of  course  zero  when  the  connecting  rod  is  rigid. 

It  thus  appears  that  the  internal  reactions  of  a  system  of 
points  connected  as  above  described,  are  eliminated  from  the 
equation  of  virtual  work  by  selecting  the  virtual  displacements 
so  as  to  leave  the  lengths  of  the  connecting  rods  or  threads 
unchanged.  This  can  always  be  done  when  the  rods  and 
threads  are  not  elastic.  When  they  are  elastic,  the  equation  of 
virtual  work  will  contain  terms  of  the  form  T§1.  These  terms 
must  then  be  determined  from  the  known  relation  between  the 
tension  and  the  length  of  an  elastic  rod  or  thread. 

248.  It  is  somewhat  difficult  to  prove  the  principle  of  virtual 
'work  for  the  most  general  case  of  any  system  of  bodies  although 
this  is  the  case  in  which  it  finds  its  most  important  application. 
It  is  evident,  however,  that  the  principle  will  be  true  in  this 
general  case  provided  that  all  the  connections  and  reactions 
between  the  different  bodies  constituting  the  system  be  ex- 
pressed by  means  of  forces  and  introduced  into  the  equation 
of  virtual  work.  The  difficulty  lies  in  expressing  the  connec- 
tions existing  between  the  parts  of  the  system  by  means  of 
forces. 

But  most  of  these  internal  reactions  can  be  shown  to  dis- 
appear from  the  equation  of  virtual  work,  so  that  they  need 
not  be  taken  into  account. 


I54  STATICS.  [249. 

Thus  the  tension  of  an  inelastic  thread  or  rod  being  com- 
posed of  two  equal  and  opposite  forces  does  no  work  in  any 
infinitesimal  displacement  ;  the  work  of  the  reaction  of  a  fixed 
point  or  fixed  axis  is  zero,  because  the  point  of  application 
cannot  move  in  the  direction  of  the  force  ;  the  normal  reaction 
of  a  surface  along  which  a  body  is  constrained  to  slide  does  no 
work,  because  any  possible  displacement  is  at  right  angles  to 
the  force.  If,  however,  the  surface  be  rough,  the  friction  will 
in  general  do  work. 

249.  A  full  discussion  of  the  principle  of  virtual   work  for 
the  most  general  case  cannot  be  given  in  this  place.     It  must 
suffice  to   state  it  and   to   illustrate  its   application    in   a  few 
special  cases. 

The  necessary  and  sufficient  condition  of  equilibrium  of  a 
system  of  bodies  the  connections  between  which  can  be  expressed 
by  forces  is  this,  that  the  sum  of  the  virtual  works  of  all  the 
forces  must  vanish  for  all  displacements  consistent  with  the 
geometrical  conditions  to  which  the  system  may  be  subject. 

The  internal  reactions  between  the  different  parts  of  the 
system,  with  the  exception  of  friction  and  elasticity,  will  in 
general  not  enter  into  the  equation  of  virtual  work.  Such 
reactions  can  therefore  be  neglected  while  friction  and  elastic 
tensions  must  be  included  among  the  forces  acting  on  the 
system. 

250.  Let  P  be  one  of  the  forces,  %p  the  projection  on  its 
direction  of  the  virtual  displacement  of  its  point  of  application  ; 
then  the  principle  of  virtual  work  requires  that 


If  X,  Y,  Z  be  the  rectangular  components  of  P,  and  x,  y,  z 
the  co-ordinates  of  its  point  of  application,  the  same  condition 
can  be  expressed  in  the  form  : 


251.]  THE   PRINCIPLE   OF   VIRTUAL   WORK.  ^5 

251.  Whether  the  system  be  in  equilibrium  or  not,  the  quan- 
tity 2/^=0  2(-£&r+  YSy  +  ZSz)  represents  the  element  of  work 
8  W  done  by  the  forces  in  a  virtual  displacement.  For  an  actual 
displacement  of  the  system  B  should  be  replaced  by  d,  and  we 
have 

Ydy+Zdz) 


for  the  work  done  in  the  infinitesimal  displacement. 

Most  of  the  forces  occurring  in  nature  are  such  as  to  make 
this  quantity  an  exact  differential.  In  this  case  the  forces  are 
said  to  form  a  conservative  system,  and  the  equation  can  be 
integrated  from  any  initial  position  or  configuration  of  the  sys- 
tem to  any  final  position  or  configuration  without  reference  to 
|the  intermediate  positions. 

Taking  the  initial  position  as  the  standard  of  reference,  the 
Iresult  can  be  written  in  the  form 


(where  U  is  a  function  of  the  co-ordinates  (or  other  quantities) 
letermining  the  position  and  configuration  of  the  system,  while 
\UQ  is  the  initial  value  of  U. 

Leaving  the  standard  of  reference  indeterminate,   the  equa- 
:ion  can  be  written  in  the  form 

w=u+c> 

\C  being  merely  a  symbol  for  the  constant  of  integration.  The 
[unction  Wis  called  the  work  function  or  force  function. 

If  the  final  position  of  the  system  be  taken  as  standard  of 
iference,'  and  £/i  be  the  value  of  U  in  the  final  position,  the 

luation  takes  the  form 


'here  V  is  called  the  potential  energy  of  the  forces  with  refer- 
ence to  the  final  position. 


156  STATICS.  [252. 

252.    If  the  work  function 

W=U+C 

be  given  as  a  known  function  of  the  co-ordinates  determining 
the  configuration  of  the  system,  the  positions  of  equilibrium  of 
the  system  can  be  found  from  the  condition 


which  expresses  that  the  work  function  W  is  a  maximum,  or  a 
minimum  (or  stationary).  It  can  be  shown  without  difficulty 
that  the  equilibrium  is  stable  when  the  work  function  is  a  maxi- 
mum, and  unstable  when  the  work  function  is  a  minimum. 

As  the  potential  energy  by  the  formulae  of  Art.  248  is  equal 
to  the  work  function,  but  of  opposite  sign,  it  follows  that 
the  equilibrium  is  stable  or  unstable  according  as  the  potential 
energy  is  a  minimum  or  a  maximum. 

253.  The  special  case  when  the  only  forces  acting  are  the  weights  of 
the  particles  constituting  the  system  is  worth  mentioning. 

Let  m  be  the  mass  of  one  of  the  particles,  mg  its  weight,  and  z  its 
height  above  a  horizontal  plane  of  reference.  Then  the  virtual  work  of 
the  weights  is 


If  z  be  the  height  of  the  centroid  of  the  system  above  the  plane  of  refer- 
ence,  we  have  ^mz  =  ^m  •  z  ;  hence  ^m  Bz  =  ^m  •  8z.  The  work  func- 
tion is,  therefore, 

W=—g$m-z+  C, 

and  this  becomes  a  maximum  or  minimum  according  as  z  is  a  minimui 
or  maximum  ;  i.e.  the  equilibrium  is  stable  or  unstable  according  as  tht 
centroid  of  the  system  is  at  its  least  or  greatest  height. 

254.  It  is  the  object  of  every  machine  to  do  work  in  a  certain  pre- 
scribed way,  i.e.  to  exert  force,  or  overcome  a  resistance,  through  a  cer- 
tain distance.  The  various  forces  of  nature,  such  as  the  muscular  foi 
of  man  and  other  animals,  the  force  of  gravity,  the  pressure  of  the  windj 
electricity,  the  expansive  force  of  steam  or  gas,  etc.,  are  called  upon  fo^ 
this  purpose.  In  most  cases  it  would  not  do  to  apply  these  forces* 


256.]  THE   PRINCIPLE    OF   VIRTUAL  WORK.  157 

directly ;  they  must  be  controlled,  guided,  and  transformed  in  various 
ways  to  become  useful,  and  this  is  done  by  interposing  the  machine 
between  the  given  driving  force,  commonly  called  the  power,  and  the 
force  which  is  to  do  the  final  work,  usually  called  the  resistance,  load,  or 
weight.  We  shall  in  general  denote  the  "  power  "  by  P,  the  "  weight  " 

by  Q. 

The  term  power  is  somewhat  objectionable  in  this  connection,  being 
here  used  to  denote  a  force,  while  in  Kinetics  it  is  used  for  the  rate  of 
doing  work. 

255.  The  ratio  Q/P  of  the  weight  to  the  power  is  called  the  mechani- 
cal advantage  of  the  machine. 

Under  the  action  of  the  power  P  its  point  of  application  as  well  as 
that  of  the  weight  Q  is  displaced.  The  corresponding  work  of  the  force 
P  may  be  called  the  available  or  total  work;  that  of  the  force  Q  is 
called  the  useful  work. 

The  ratio  of  the  useful  work  to  the  total  work  is  called  the  efficiency 
of  the  machine. 

In  all  machines  this  efficiency  is  a  proper  fraction,  owing  to  the  fact 
that  the  work  done  by  P  must  balance  not  only  the  useful  work,  but  also 
the  so-called  wasteful  work  due  to  friction,  stiffness  of  ropes,  slipping  of 
belts,  lack  of  rigidity,  etc. 

256.  For  a  more  complete  discussion  of  the  principle  of  virtual  work 
the  student  is  referred  to  MINCHIN'S  Statics,  Vol.  I.,  pp.  78-96,  160-180, 
and  Vol.   II.,  pp.  98-188;    ROUTH'S   Statics,  Vol.  I.,  pp.    146-197; 
SCHELL'S  Theorie  der  Bewegung  und  der  Kr'dfte,  Vol.  II.,  pp.  166-211  ; 
and  J.  PETERSEN,  Statik  fester  Korper,  iibersetzt  von  R.  von  Fischer- 
Benzon,  Kopenhagen,  Host,  1882,  pp.  114-124. 


158  STATICS.  [257. 

VIII.    Theory  of  Attractive  Forces. 

I.     ATTRACTION. 

257.  Among  the  various  kinds  of  forces  introduced  in  physics 
for  describing  and  interpreting  natural  phenomena,  forces  of 
attraction  and  repulsion  occupy  a  most  prominent  place. 

According  to  Neivtons  law  (the  law  of  universal  or  cosmical 
gravitation,  the  law  of  nature),  every  particle  of  matter  attracts 
every  other  such  particle  with  a  force  proportional  to  the  masses 
and  inversely  proportional  to  the  square  of  the  distance  of  the| 
particles. 

If  m,  m1  be  the  masses  of  the  two  particles,  r  their  distance, 
and  K  a  constant,  the  mathematical  expression  for  the  force  oH 
mutual  attraction  exerted  by  each  particle  on  the  other,  or  for 
the  stress  between  them,  is,  therefore, 

*=**£.  (i; 


258.  Each  particle  is  here  regarded  as  a  mathematk 
point  at  which  its  mass  is  concentrated.  The  attractive  forc< 
would,  therefore,  approach  the  limit  oo  as  the  distance  betweei 
the  points  approaches  the  value  o.  To  prevent  the  introductioi 
of  infinite  forces,  we  may  in  such  limiting  cases  regard  the  par- 
ticles as  very  small  homogeneous  spheres  formed  of  an  impene- 
trable substance.  If  r,  r*  be  the  radii,  p,  p1  the  densities  of  th< 
spheres,  the  attraction  reaches  a  finite  maximum  value 
when  the  spheres  are  in  contact,  viz. 

tw" 


which  is  very  small  of  the  fourth  order  if  r,  r1  be  very  small  of 
the  first  order.     Thus,  for  r=rl, 


26i.]  ATTRACTION.  ^9, 

259.  In  many  applications  of  the  theory  of  attraction,  in  par- 
ticular in  electricity  and  magnetism,  it  is  convenient  to  consider 
forces  of  repulsion.     This  only  requires  a  change  of  sign  in  the 
expression  for  the   force,  and  this   sign   may  be    regarded   as 
attaching  to  the  mass  of  one  of  the  particles  ;  in  other  words, 
if  the  mass  of  a  centre  of  attraction  be  taken  as  positive,  that  of 
a  centre  of  repulsion  is  taken  to  be  negative,  or  vice  versa. 

260.  While   according    to    Newton's    law    (i)    the    force   is 
inversely  proportional  to  the  square  of  the  distance,  it  is  often 
convenient  to  use  forces  depending  upon  the  distance  r  in   a 
different  way.     Thus  the  theory  of  Newtonian  attraction  can  be 
generalized  by  assuming  for  the  force  between  two  particles  my 
m'  the  law 

F=Kmm'f(r),  (2) 

where /(r)  represents  any  function  of  the  distance  r. 

When  nothing  is  said  to  the  contrary,  we  shall  here  always 
assume  that/(r)  =  i/^2,  as  in  Newton's  law  (i). 

261.  The  constant  K  evidently  represents  the  force  with  which 
two  particles,  each  of  mass    i,  attract  each  other  when  at  the 
distance    i.     It  is   a   physical    constant   to   be  determined  by 
experiment,  and  its  numerical  value  depends  on  the  units  of 
measurement  adopted.     What  can    be  directly  observed  is  of 
course  not  the  force  itself,  but  the  acceleration   it  produces. 
Dividing  the  force  F,  as  given  by  formula  (i),  by  the  mass  m' 
of  the  particle  on  which  it  acts,  we  find  for  the  acceleration  j 
produced  by  the  attraction  of  the  mass  m  in  the  mass  m1  at  the 
distance  r  from  m  : 

m  /  \ 

7=«-2-  (3) 

This  quantity  may  also  be  regarded  as  the  force  of  attraction 
exerted  by  the  mass  m  on  a  mass  i  at  the  distance  r  from  m, 
and  is  therefore  called  briefly  the  attraction  at  the  point  where 
the  mass  i  is  situated. 


!6o  STATICS.  [262. 

262.  It  will  be  shown  later  (Art.  273)  that  the  attraction  of 
.a  homogeneous  sphere  on*  an  external  point  is  the  same  as  if 
the  mass  of  the  sphere  were  concentrated  at  its  centre.  Thus, 
if  m  be  the  mass  of.  the  earth  (here  assumed  as  a  homogeneous 
sphere),  the  attraction  it  exerts  on  a  mass  I  situated  at  a  point 
P  above  its  surface,  at  the  distance  OP  =  r  from  the  centre  O,  is 
^Km/r*',  and  this  is  also  the  acceleration  j  that  it  would  cause 
in  any  mass  m'  at  P. 

Now  for  points  P  near  the  earth's  surface  this  acceleration  j 
is  known  from  experiments;  it  is  the  acceleration  of  gravity, 
usually  denoted  by  g.  As  the  radius  of  the  earth,  ^=6.37  x  io8 
centimetres,  and  its  mean  density  /o  =  5f,  are  also  known,  the 

value  of  the  constant  K  can  be  found  from  the  formula 

i 

m 


or  *:= 

With  £-=980  we  find  in  C.G.S.  units 

K  =  -  -  -  -  =  o.ooo  ooo  0648. 
1.543*  I0' 

This,  then,  is  the  force  in  dynes  with  which  two  masses  of  i 
gramme  each  would  attract  each  other  if  concentrated  at  two 
points  i  centimetre  apart. 

263.  Exercises. 

(1)  Show  that  the  value  of  K  in  the  F.P.S.  system  is  -  - 

9.8  x  io8 

(2)  When  the  units  are  so  selected  as  to  make  the  constant  K  equa 
Tto  i,  they  are  called  astronomical  units.     Show  that  the  astronomica 

unit  of  mass,  i.e.  a  mass  which  when  concentrated  at  a  point  produces 
unit  acceleration  at  unit  distance,  is=  I/K. 

264.  Let  a  mass  or  a  system  of  masses  be  given,  and  let  it  be 
required  to  determine  the  attraction  at  any  point  P  (Art.  261) 
produced  by  it.      The  given  masses  may  consist  of  discrete  par- 
ticles, or  they  may  be  continuous  of  one,  two,  or  three  dimen- 
sions.    Continuous  masses  must  be  resolved  into  elernents;  the 


j 


26S.] 


ATTRACTION. 


161 


attraction  at  P  produced  by  each  element  must  be  determined, 
and  then  all  these  forces  must  be  compounded  into  a  single 
resultant.  This  is  always  possible  because  the  forces  all  pass 
through  the  point  P. 

Let  dm  be  the  element  of  mass  situated  at  the  point  Q,  P  the 
attracted  point  of  mass  i,  PQ  =  rihe  distance  between  them, 
and  a,  ft,  7  the  direction  cosines  of  r\  then  the  attraction  at  P 
due  to  dm  is  dm/r^\  its  components  are  adm/r1,  ftdm/fl, 
and  the  components  of  the  resultant  attraction  R  at  P  are 


<•>..  Y=S>  z=$' 


(4) 


where  the  integrations  must  be  extended  over  the  whole  given 
mass.  The  resultant  R  .itself,  and  its  direction  cosines  a,  b,  c, 
are  finally  found  from  the  formulae 

V  V  "7 

•**•        7.          •*  *" 

a=R'  b=R'  C=ll 


(5) 


The  following  examples  will  illustrate  the  process. 


265.  Homogeneous  Circular  Arc.  To  determine  the  attraction 
•exerted  by  a  mass  distributed  uniformly  along  a  circular  arc  ACB 
(Fig.  78)  of  angle  2  a  and  radius  a  on  a  mass  i  situated  at  the  centre 
Pof  the  circle,  let  QQ  =  ds  be  an  element  of  the 
arc,  dm  =  pds  its  mass  ;  then 


is  its  attraction  at  P,  and  this  force  has  the  direc- 
tion PQ. 

Resolving  this  force  parallel  to  the  bisecting 
radius  PC  of  the  arc  and  at  right  angles  to  it,  it 
will  be  seen  that  the  latter  component  need  not  be 
considered,  since  it  is  balanced  by  an  equal  and 
opposite  component  arising  from  the  element  sit- 
uated symmetrically  to  QQ  with  respect  to  PC. 
If  K  CPQ  =  <9,  the  component  along  PC  is 

ds  cos  Q/a2,  or  since  ds  —  adO,  Kp  cos  Odd /a.      Hence  the  resultant 
attraction  at  P  is  • 

J?  —  KP  C+a       (Mti  —         sin  ft 
a  J —a.  Q 

PART   II — II 


(6) 


1 62 


STATICS. 


[266. 


Denoting  the  chord  AB  of  the  given  arc  by  c,  the  result  can  be  put  into 
the  form 

R  =  *?•-<,  (6f) 

which  might  have  been  found  directly  from  Fig.  78,  without  integration, 
since  ds  cos  0  =  qq\ 

Formula  (6f)  shows  that,  if  the  chord  AB  were  covered  with  mass  of 
the  same  density  p  as  the  arc,  and  if  this  mass  were  concentrated  at  the 
middle  point  C  of  the  arc,  it  would  produce  at  P  the  same  attraction  as 
the  axcACB. 

266.  Homogeneous  Straight  Rod.  To  determine  the  attraction  at 
any  point  P  produced  by  a  mass  distributed  uniformly  along  a  segment  of 
a  straight  line,  AB,  let  QQ'  =  ds  (Fig.  79)  be  an 
element,  OQ  =  s  its  distance  from  the  foot  of  the 
perpendicular  PO=p  dropped  from  P  on  AB, 
and  let  r,  0  be  the  polar  co-ordinates  of  Q  with 
respect  to  P  as  pole  and  PO  as  polar  axis.  Re- 
solving the  attraction  Kpds/r*  of  the  element  QQ1 
along  PO  and  at  right  angles  to  it,  we  find  the 
components 


,v 

a  A  •=. 


cos  Oils 

—  —  —  f 


sin  Oils 


Fig.  79. 


The  figure  gives  r=//cos0,  j=/tan0;  hence 
;  substituting  these  values,  we  have 


/  / 

and  integrating  between  the  limits  —  ft  =  OPB  and  a  =  OPA  : 


X=  5£  (sin/?  +  sin  a),     Y=  &  (cos/8  -  cosa). 
/  P 

The  resultant  attraction 


(7) 


makes  with  PO  an  angle  $,  for  which  we  have 


X      sin/5+smce 
hence 


268.]  ATTRACTION.  163 

n^;  (8) 

2 

'     (9) 
i.e.  the  attraction  at  P  bisects  the  angle  APB  subtended  at  P  by  the  rod. 

267.  These  results  might  have  been  derived  from  the  problem  of 
Art.  265.  For  it  is  easy  to  see  that  the  attraction  exerted  at  P  by  the 
straight  rod  AB  is  the  same  as  that  exerted  by  the  circular  arc  ab  (Fig, 
79)  described  about  P  with  radius/  and  bounded  by  PA,  PB.  This  will 
follow  if  it  can  be  shown  that  the  attraction  at  P  of  the  element  QQ1' 
of  the  rod  is  equal  to  that  of  the  element  qq*  of  the  arc  contained 
between  the  same  radii  vectores.  Now  the  attraction  of  QQ'  is 
KP'QQ7r~>  while  that  of  W*  is  KP  '$$'//'  Projecting  QQ'  on  the 
circle  of  radius  r,  we  have 

n'     _p 

QQ'cosO~~r 

or  since  the  triangle  POQ  gives  cosO=fl/r, 


which  proves  the  proposition. 

268.  It  has  been  shown  in  Art.  266  that  the  attraction  at  any  point 
P  exerted  by  a  straight  rod  AB  bisects  the  angle  APB  ;  it  is  therefore 
tangent  to  the  hyperbola  passing  through  P  and  having  A,  B  as  its  foci. 
Hence  if  in  any  plane  through  AB  the  system  of  confocal  hyperbolas 
be  constructed  with  A,  B  as  foci,  the  direction  of  the  attraction  at  any 
point  P  in  the  plane  is  along  the  tangent  to  the  hyperbola  that  passes 
through  P.  These  hyperbolas  having  everywhere  the  direction  of  the 
resulting  attractive  force,  are  called  the  lines  of  force. 

An  ellipse  passing  through  P  and  having  the  same  foci  A,  B  would 
have  the  bisector  of  the  angle  APB  as  its  normal.  The  confocal 
ellipses  about  A,  B  as  foci  form  the  so-called  orthogonal  system  of  the 
lines  of  force.  If  such  an  ellipse  be  regarded  as  offering  a  normal 
resistance,  the  point  P  would  be  kept  in  equilibrium  under  the  action 
of  the  attraction  of  the  rod  and  the  reaction  of  the  curve.  The  con- 
focal  ellipses  are  therefore  called  equilibrium,  or  level,  lines,  or  also 
for  a  reason  that  will  appear  later  equipotential  lines. 


164  STATICS.  [269. 

Rotating  the  whole  figure  about  AB  as  axis,  the  ellipses  describe  con- 
focal  ellipsoids  of  revolution  which  are  level,  or  equipotential,  surfaces. 

269.  Exercises. 

(1)  A  segment  AB  is  cut  out  of  an  infinite  straight  line  along  which 
mass  is  distributed  uniformly.     If  the  mass  on  the  ray  issuing  from  A  be- 
repulsive,  that  on  the  ray  issuing  from  B  (in  the  opposite  sense),  attrac- 
tive, determine  the  resultant  attraction  at  any  point  P  by  the  method  of 
Art.  267,  and  show  that  the  lines  of  force  are  confocal  ellipses,  while  the 
equipotential  surfaces  are  confocal  hyperboloids. 

(2)  Three  rods  of  constant  density  form  a  triangle.     Find  the  point 
at  which  the  resultant  attraction  is  zero. 

(3)  Find  the  attraction  of  a  straight  rod  AB  of  constant  density  on 
a  point  /'situated  on  the  line  AB  so  that  AP=  a,  J3P  =  b. 

(4)  Two  rods  of  lengths  2  a,  ib,  and  of  equal  constant  density,  are 
placed  parallel  to  each  other,  at  a  distance  c,  so  that  the  line  joining 
their  middle  points  is  at  right  angles  to  them.    Find  their  mutual  attrac- 
tion, i.e.  the  force  required  to  keep  them  apart. 

(5)  Show  that  the  attraction  of  a  homogeneous  rod  of  infinite  length 
on  a  point  at  the  distance  p  from  it,  is  2  *p/p. 

270.  The  formula  of  the  last  exercise  (5)  can  be  used  to  determine 
the  attraction  of  an  infinitely  long  homogeneous  cylinder  of  finite  cross- 
section  on  an  external  point  P,  by  resolving  the  cylinder  into  filaments 


N 


Fig.  80. 

parallel  to  the  axis.  Fig.  80  represents  the  cross-section  of  the  cylinder 
passing  through  P.  The  polar  element  of  area  at  Q,  rdd  •  dr,  can  be 
regarded  as  the  cross-section  of  a  filament,  whose  attraction  at  P  is,  by 

Ex.  (5), 

rdrdB  j    ,/\ 

2  Kp =  2  Kparati. 


272.]  ATTRACTION.  !65 

Resolving  this  force  along  and  at  right  angles  to  PO,  and  considering 
that  owing  to  symmetry  the  resultant  R  must  pass  through  O,  we  have 


If  the  radius  vector  PQ  meet  the  surface  of  the  cylinder  at  M  and  N, 
the  integration  with  respect  to  r  gives 


With  PO  =  p,  we  have  MN—  2vV—  /2  sin20,  and  the  limits  of  the 
integration  are  ±  sin"1  (#//).     Hence 


271.  Homogeneous  Circular  Plate.  To  determine  the  attraction  of 
a  homogeneous  circular  area  on  a  point  P  situated  on  the  axis  through 
the  centre  O  at  right  angles  to  its  plane  at  the  distance  PO=p,  we 
may  resolve  the  plate  into  ring-shaped  elements  of  radii  r  and  r  -f-  dr. 
The  mass  of  such  a  ring  is  2  irprdr  ;  all  points  of  the  ring  are  at  the 
same  distance  Vr2  +/2  from  P,  and  their  attractions  make  the  same 
angle  <£  =  \axrl(r/p)  with  the  axis  PO.  Hence  the  attraction  of  the 
ring  is 

cos  <f>  =  2  TTKp  sin  <pd<pj 


since  dr  =/^>/cos2  <£  =  (/2  + 

Let  2  «  be  the  vertical  angle  of  the  cone  that  subtends  the  plate  at 
P;  then 

/»<x  a 

^?  =   2  7TK/3    I       Sin  <£</<£  =  4  TTKp  SUi*-,  (l  l) 

or,  in  terms  of/, 

R=2TTKp(l--—^^\  (ll') 

V  V^  +// 

272.  Homogeneous  Spherical  Shell  :  Geometrical  Method,  (a)  Attrac- 
tion at  an  internal  point  P.  Let  C  be  the  centre,  a  the  radius  of  the 
sphere  (Fig.  81).  A  thin  double  cone  having  its  vertex  at  Pcuts  the 
sphere  in  two  elements,  AB  =  dS,  A'£'  =  dS1,  which  can  be  shown  to 
exert  equal  and  opposite  attractions  at  P. 


1 66 


STATICS. 


[273- 


Let  PA  =  r,  PA1  =  r',  and  let  ^/<o  denote  the  solid  angle  of  the  cone 
(i.e.  the  area  it  cuts  out  of  a  sphere  of  radius  i  described  about  P  as 
centre)  ;  then-rv/w  is  the  area  cut  out  of  a 
sphere  of  radius  r  with  the  same  centre 
P.  Hence  the  element  of  mass  at  A  is 
p  •  ^Vo>/cos  PA  C,  and  its  attraction  at  P  is 
=  Kpdw/cosPAC.  Similarly,  the  attraction 
of  the  mass  element  at  A'  is 

=  KPr'-do>/(r''2cos  PA'C)  =  xprfu/cosPA'C. 

These  attractions  are  equal,  since  for  the 
sphere  %PAC=^PA'C. 

The  whole  sphere  can  thus  be  divided  up 
into  elements  exerting  equal  and  opposite  attractions  at  P  ;  the  resultant 
attraction  of  the  whole  shell  at  any  internal  point  is,  therefore,  zero. 

273.  (b)  Attraction  at  an  external  point  P.  The  investigation  can 
be  made  similar  to  that  for  an  internal  point  by  introducing  the  point 
P1  (Fig.  82),  which  is  inverse  to  /*with  respect  to  the  sphere,  i.e.  the 
point  P'  on  CP  for  which  CP-  CP'  =  CA\  or  putting  CP=p,  CP'=p', 
CA  =  a,  the  point  for  which 

pp'  =  a\  (12) 

Any  chord  HH*  through  P'  determines  two  pairs  of  similar  triangles  : 
CUP1  and  CPH,  CJf'P'  and  CPH* ;  for  each  pair  has  the  angle  at  C 


Fig.  82. 

in  common,  and  the  sides  including  the  equal  angles  proportional  by 
(12-),  since  CH=  CH'=a.  It  follows  that  £  C&P'  =  K  CPH,  and 
^  CH'P'  =  %.  CPU';  hence,  as  the  triangle  HCH'  is  isosceles,  the  line 
CP  bisects  the  angle  HPH' . 

With  the  aid  of  these  geometrical  properties  it  can  be  shown  that 


274-] 


ATTRACTION. 


equal  attractions  are  produced  at  P  by  the  elements  dS  at  H  and  dS'  at 
//'  cut  out  by  any  thin  cone  whose  vertex  is  the  inverse  point  P'.  We 
have,  as  in  Art.  272,  for  the  mass  elements  at  H  and  H1  cut  out  by  the 

COIle  l  J 


and  for  the  corresponding  attractions  at 


but  these   expressions   are   equal   since   £  /"77C  =  £  P'ff'C  and  the 
similar  triangles  give  r/PH=  a/p,  r'/PH'  =  a  /p. 

As,  moreover,  these  attractions  make  equal  angles  with   CPt  their 
projections  on  this  line  are  equal,  and  their  resultant  is 


— 

To  form  the  final  resultant,  this  expression  must  be  integrated  over 
the  whole  sphere,  and  as  the  summation  of  the  double  cone  gives 
I  //a  =  2  TT,  we  find 

D 
R= 


where  M  denotes  the  whole  mass  of  the  shell.  Hence,  the  attraction 
of  a  homogeneous  shell  on  an  external  point  is  the  same  as  if  the  whole 
mass  of  the  shell  were  concentrated  at  the  centre  of  the  shell. 

274.  (c)  Attraction  at  the  surface.  If  the  point  P  approaches  the 
surface  from  within,  the  attraction  remains  constantly  zero  ;  if  P 
approaches  the  surface  from  without,  the  at- 
traction KM/p2  approaches  the  limit  nM/a*. 
For  a  point  on  the  surface  the  attraction  is 
the  arithmetic  mean  of  these  two  values,  viz. 

R=   27TKp.  (l4) 

This  can  be  shown  as  follows  (Fig.  83). 
The  element  of  mass  at  H  is 


pdS=p-  rWa>/cos  PHC  ; 


Fig.  83. 


its  attraction  at  P  is  Kp  dw/cos  PHC,  and  as  the  angles  at  P  and  H  are 
equal,  the  projection  of  the  attraction  on  PC  =  Kprfv.  For  a  point 
on  the  surface  \dw=  271-.  Hence  the  total  attraction  at  /Ms  =  27r/c/o. 


1 68 


STATICS. 


[275- 


275.    Homogeneous  Spherical  Shell :     Analytical  method.     Let   Q 
(Fig.  84)   be  any  point  on  the  sphere;   PQ  =  r,    CQ—a,   CP=p, 

^-PCQ  =  0.  Through  Q  lay 
a  plane  at  right  angles  to  CP, 
and  take  as  element  the  mass 
contained  between  this  and  an 
infinitely  near  parallel  plane. 
P  This  mass  element  is 

=  p  •  2  ira  sin  0  *  adO, 

and  its   attraction   at  P  along 
CPis 


Fig.  84. 


a2  sin  OdO-±>  cos  CPQ  = 


sin  OdO  . 


P  ~  a 


The  relation  between  r  and  0  follows  from  the  triangle  CPQ,  which 
gives 


—zap  cos  0, 


hence 


rdr  =  ap  sin 


Substituting  for  a  sin  OdO  and  for  a  cos  0  their  values  from  the  last 
two  relations,  the  expression  for  the  attraction  of  the  elementary  ring. 
becomes 


rdr 
2VKpa.-- 


a     fi-cf+r>     , 
=  „,_.<     —^     -.dr. 


(a)  For  an  internal  point  P,  we  have  p<a,  and  the  limits  for  r  are 
from  a  —p  to  a  +p.     Hence  the  resultant  attraction  is 


=  o. 


(b)  For  an  external  point  P,  we  have  p>a,  and  the  limits  are  from 
p  —  a  to  p  +  a.     Hence  the  attraction  becomes 


* 

R  = 


]*>+<* 
,-.= 


M 


1  £. 

'A. 

276.   Exercises. 

(i)  Show  that  the  attraction  exerted  by  a  right  circular  cone  of  ver- 
tical angle  2  a  and  height  h,  at  the  vertex,  is  =  2  TTK/O  ( i  — 


277.]  POTENTIAL. 

(2)  Show  that  the  attraction  of  a  circular  cylinder  of  radius  a  and  of 
length  /,  at  a  point  on  its  axis  at  the  distance  x  from  the  nearest  base,  is 


(3)  From  the  result  of  Ex.  (2)  show  that  the  attraction  of  a  cylinder 
extending  in  one  sense  to  infinity  is  =  2  TTKpa  at  its  base. 

(4)  Show  that  for  a  spherical  shell  of  finite  thickness,  if  the  density 
be  either  constant  or  a  function  of  the  distance  from  the  centre  only, 
the  attraction  is  zero  at  any  point  within  the  hollow  of  the  shell,  and 
that  it  is  the  same  as  if  the  whole  mass  were  concentrated  at  the  centre 
at  any  external  point. 

(5)  Show  how  to  find  the  attraction  of  a  homogeneous  spherical  shell 
of  finite  thickness,  at  any  point  within  the  mass  of  the  shell. 

(6)  Show  that  the  attraction  of  a  solid  sphere  of  mass  Mt  the  density 
being  any  function  of  the  distance  from  the  centre,  is  =  AcJ///2  at  any 
external  point  P,  having  the  distance  /  from  the  centre,  and  that  it  is 
directly  proportional  to  the  distance  of  the  point  P  from  the  centre 
when  P  lies  within  the  mass. 

(7)  Show  that  the  attraction  of  a  solid  homogeneous  hemisphere  at  a 
point  in  its  edge  is  =  f  K/o^V^-f  4,  and  that  it  makes  with  the  plane 
of  the  base  an  angle  of  about  32^°. 

2.     THE    POTENTIAL. 

277.  The  configuration  and  density  of  any  attracting  masses 
being  given,  the  force  of  attraction  R  exerted  by  these  masses 
on  a  mass  I  situated  at  any  point  P  can  be  determined  both  in 
magnitude  and  direction.  The  method  illustrated  on  some  sim- 
ple examples  in  the  preceding  articles,  while  theoretically  quite 
general,  becomes  very  laborious  in  more  complicated  cases. 
Moreover,  the  required  resultant  R,  i.e.  the  "attraction  at  the 
point  P,"  depends  as  to  its  magnitude  and  direction  on  the 
position  of  the  point  P  ;  and  it  is  of  interest  to  investigate  its 
variation  from  point  to  point  throughout  space,  in  a  similar  way 
as  was  done  for  the  example  of  the  straight  rod  in  Art.  268. 

Investigations  of  tbis  kind  are  greatly  facilitated  by  the  aid 
of  a  certain  function  called  the  potential,  whose  meaning  and 
use  we  proceed  to  discuss  very  briefly  in  the  following  articles. 


170 


STATICS. 


[278. 


Fig.  85. 


278.  The  attraction  at  the  point  P  exerted  by  a  single  particle 
Q  of  mass  m  (Fig.  85)  is  =m/rz  if  the  units  be  so  selected  as 

to  make  the  constant  K=  i  (Art. 
262  and  Art.  263,  Ex.  2).  (This 
assumption  is  generally  made 
in  theoretical  investigations,  as 
there  is  nothing  to  be  gained  by 
carrying  the  constant  factor  K 
through  all  the  formulae.  The 
factor  K  can  always  be  re-intro- 
duced when  numerical  results 
are  required.) 

Let  the  particle  P  be  displaced  through  an  infinitesimal  dis- 
tance PP'  =  ds  in  any  direction,  and  let  <f>  be  the  angle  between 
QP  =  r  and  PP'.  The  element  of  work  done  by  the  force  m/fl 
in  this  displacement  is 

jji/        m          ij          m  dr  ,       d(ni 

aW=-  —  cos$ds= —  ds  =  —  (  — 

1T  r*>  ds          ds\r 

The  quantity  m/r  occurring  in  the  last  expression  is  called  the 
potential  of  the  mass  m  at  the  point  P]  it  is  usually  denoted 
by  V. 

279.  If  the  particle  continue  to  move  along  some  curve  from 
its  initial  position  P  to  some  final  position  Pv  the  total  work 
done  by  the  attraction  of  Q  is  evidently 


TTT  J  T  7"  TT 

where  F=7/2/ris  the  potential  at  P,  and  V^  =  m/r^  is  the  poten- 
tial at  Pv  Hence,  the  difference  of  the  potentials  at  any  two 
points  is  equal  to  the  work  done  by  the  attraction,  whatever  may 
have  been  the  path  along  which  the  displacement  has  taken 
place. 

As  the  potential  V=m/r  becomes  zero  when  r=oo,  it  appears N 
that  the  potential  V1  at  any  point  Px  is  the  work  that  would  be 


28i.]  POTENTIAL.  l-Jl 

done  by  the  attraction  on  a  particle  of  mass  I  if  it  were  brought 
up  to  the  point  PJ  along  any  path  from  infinity. 

The  relations  of  Art.  278  can  be  written  in  the  form 

dV        m 

— =  --cos</>; 

ds         r* 

i.e.  the  derivative  of  the  potential  with  respect  to  any  displacement 
is  equal  to  the  component  of  the  attraction  in  the  direction  of  the 
displacement. 

280.  When  there  are  given  several  masses  m,  m\  mn , ...,  con- 
centrated at  points  Q,  Qf,  Q",  ...,  their  potential  at  any  point  P 
is  defined  as  the  sum 

jr_m     m'     mn         _^m 

v  — — I — -f  H — 77  ~r"*  —  Z —  > 
r      r       r"  r 

when  the  given  masses  are  continuous,  the  sign  of  summation 
must  be  replaced  by  an  integral,  and  we  have 


The  fundamental  properties  proved  in  Art.  279  remain  the 
same. 

281.  Let  there  be  given  a  continuous  mass  m,  referred  to 
a  rectangular  system  of  co-ordinates.  The  attraction  at  any 
point  P  (x,  y,  z)  due  to  this  mass  has  three  components  X,  F,  Z, 
which  can  be  found  as  follows.  The  attraction  produced  at  P 
by  an  element  dm  at  a  point  Q  (x't  y'  ,  z')  of  the  mass  is  dm/r2, 
where  r=PQ,  and  its  direction  cosines  are  (x'  —  x)/r,  (y*  —  y)/r, 
{z'  —z)/r;  hence  its  components  are 


Integrating,  we  find  the  components  of  the  total  attraction 
a.tP: 

\z'-_z)dm 
r" 


=  C 

*J 


172  STATICS.  [282. 

Differentiating  the  relation 

r*=  (x1  —x)'i-\-(y1  —y)*'  +  (z'—z)t*t 
partially  with  respect  to  x,  y,  and  z,  we  have 

dr _  _  ,    ,  _    .         dr_  _/    /_   \         dr _         ,        . 

Substituting  these  values  in  the  above  integrals,  we  find 

v          C  I  dr  j         C  &  fl\  j         d    Cdm 
JL  =  —  I  — — am=  I  —  -  }am  =  —  I  — > 

J  r*dx  J  dx\rj  dx^     r 

and  similarly 

Y=—  C^m      Z=  —  C^m 

dyJ    r  dz*s     r 

As  J  —  is  the  potential  Fof  the  given  mass,  we  have 

X= — >     Y= — >     Z 

dx  By 

i.e.  the  components  of  the  attraction  at  any  point  are  the  deriv- 
atives of  the  potential  at  that  point  in  the  direction  of  these  com- 
ponents. This  may  be  regarded  as  a  special  case  of  the  last 
proposition  of  Art.  279. 

282.  It  is  to  be  noticed  that  the  proof  given  in  the  preceding 
article  can  easily  be  extended  to  the  case  of  forces  of  the  form 
(2),  Art.  260.  In  other  words,  even  in  the  case  of  forces  not  fol- 
lowing the  Newtonian  law  of  the  inverse  square,  but  expressed 
by  any  function  f(r)  of  the  distance,  there  exists  a  function 
corresponding  to  the  potential  of  Newtonian  attractions ;  it  is 
called  \he  force  function. 

We  have,  just  as  in  Art.  281, 


hence     -= 


_,    - 
dx         r  dy         r  dz 


283.]  POTENTIAL.  I73 

These  are  the  direction  cosines  of  the  force  f(r)  with  which  the 
mass  m  at  Q(x'  r,  yl  ',  2')  attracts  the  mass  I  at  P(x,  y,  z).  The 
components  of  this  force  are,  therefore, 


,  , 

dx  dy  d 

These  expressions  show  that  there  exists  a  function 


of  which  the  components  of  the  force  at  P  are  the  partial  deriv- 
atives : 

X- 


283.    The  potential 


at  a  point  P  for  a  given  system  of  masses  is  a  function  of  the 
co-ordinates  of  the  point  P.  ,  If  this  function  be  known,  the 
attraction  at  any  point  P  produced  by  the  given  masses  can  at 
once  be  found ;  for  the  components  of  this  attraction  are 

v     dV     T/dF      7    dV 

^l  =  -,        Jr  = ,        /£  = • 

dx  dy  dz 

Hence, 

dV=Xdx+  Ydy  +  Zdz. 

If  the  function  V  be  equated  to  any  constant  F1?  the  result- 
ing equation 

v=vl 

represents  a  surface  that  is  the  focus  of  all  points  at  which  the 
potential  of  the  given  masses  has  one  and  the  same  value  Fr 
Such  a  surface  is  called  an  equipotential  surface,  or  a  level,  or 
equilibrium  smface. 


STATICS.  „       [28j. 

284.  Differentiating  the  equation  of  the  equipotential  surface, 
and  dividing  by  ds  and  by  the  attraction  R  whose  components 
are  X,  Y,  Z,  we  find 

dF  dV  87 

dx    d,r      dy     dy      dz    dz 
~R'ds+^R"dsJr~R'lds  =  0' 

The  first  factors  in  each  term  are  the  direction  cosines  of  R, 
the  second  factors  are  those  of  a  tangent  to  the  surface  ;  the 
equation  expresses,  therefore,  the  fact  that  the  attraction  R  at 
any  point  of  an  equipotential  surface  is  normal  to  the  surface. 

The  attraction  at  any  point  P  of  an  equipotential  surface 
is,  therefore,  equal  to  dVj  '  dn,  where  dn  is  the  element  of  the 
normal  at  P  between  this  and  the  next  equipotential  surface. 
Consequently,  the  attraction  is  inversely  proportional  to  the 
distance  between  the  successive  equipotential  surfaces. 

Let  the  normal  of  an  equipotential  surface  at  any  point  P  inter- 
sect the  next  equipotential  surface  at  a  point  P'  ;  let  the  normal 
at  P1  intersect  the  next  surface  at  P"  ;  and  so  on.  The  elements 
PP't  P'P",  etc.,  will  form  a  curve  which  is  at  every  point  normal 
to  the  equipotential  surface  passing  through  that  point.  Such 
a  curve  is  called  a  line  of  force,  since  its  tangent  at  any  point 
indicates  the  direction  of  the  resultant  attraction  at  that  point. 
The  lines  of  force  cut  the  equipotential  surfaces  orthogonally. 

285.  Potential  of  a  Homogeneous  Spherical  Shell.      (a)  For  an 
internal  point,  we  may  proceed  similarly  as  in  Art.  272.     The  element 
of  mass  cut  out  at  A  (Fig.  81)  by  the  small  cone  whose  solid  angle  is 
d<*  is  again  p-rVw/cosK  if  ^.PAC=  "4.PA'C  =  a  ;  the  corresponding 
potential  at  P  is  Kpn/co/cos  a  :  similarly,  the  potential  due  to  the  mass  at 
A  is  K/arVco/cos  «.     Their  sum  is 

'     3 

ait)  =  2  K 


cos  a 
since  r  -f  r*  =  2  a  cos  «. 

As  J  c/w  =  2  TT,  we  find  V\  = 

i.e.  the  potential  has  the  same  constant  value  for  all  points  within  the 
hollow  of  the  shell.  It  follows  that  the  attraction  is  zero,  as  found  in 
Art.  272. 


285.]  POTENTIAL.  Ij$ 

(^)  For  an  external  point  P,  we  might  also  proceed  geometrically, 
making  use  of  the  inverse  point  /",  as  in  Art.  273.  But  we  shall  use  the 
analytical  method. 

Just  as  in  Art.  275,  Fig.  84,  we  have  for  the  mass  of  the  ring-shaped 
element  dm  =  p  •  2  ira'2  sin  Odd,  or  as  ap  sin  BdB  =  rdr,  dm  =  2  Trpardr/p^ 
Hence  the  element  of  potential  is  dV  =  2-jrKpadr/p,  which  integrated 
between  the  limits  from  p  —  a  to  /  +  a,  gives 

M 


Hence  the  potential  is  the  same  as  if  the  whole  mass  were  concentrated 
at  the  centre. 


ANSWERS. 


Pages  29-33. 

(4)  ^a,  where  a  is  the  side  of  the  hexagon. 

(5)  At  the  centre  of  the  incircle  of  the  triangle  formed  by  the  mid- 
points of  the  sides. 

(7)  x=y  =  -  r. 

TT 

(8)  Taking  OA  as  axis  of  x, 

-  ry    y  _  2    ^" 

x  =  —  («  sin  a  +  cos  a  —  i  )  ,     y  =  —  -  (sin  a  —  a  cos  a)  . 
a2  or 


(9)     ;;=_ 

V2-f  log(l+V2)       4 

(10)  *  =  7T0,      J=|«. 

(11)  x=y  =  ±a. 


_  ^  .          _ 

=  0,     j^=  --  hijF,  where  s  =  <:(<?<>  —  e  c  )  . 
sin  0    -  i  —  cos  0     - 


(15)   With  A  as  origin,  ^  = 

-    _,(a  +  a 
=* 


.      -      ,  a  . 

first  approximation  ^  =  £  -  —  =  4.02  in.  ; 


second  approximation  x  =  \—  --=4. 50 in. 

a  -f-  b 


PART   II — 12  177 


178  ANSWERS. 

,aa+  26'   —  (2a  —  b  +  £' 


=4-5 


o.33 a>    0.25  <r. 
(*5)  *  =  - 


(26)  For  a  segment  of  a  ring  of  angle  2  a  and  radii  r1?  r2,  the  dis- 

tance of  the  centroid  from  the  centre  is  x  =  f  •  —  .  ^-±^A±^_2. 

a  n  +  >2 

Hence  ^  =  —  ^—  (740  +  73  Vi)  =  3.22  ft.  ;  i.e.  the  centroid  lies  about 

I477T 

8^  in.  above  the  door. 

(27)  *  =  $*,   J  =  f> 

(28)  ^  =    ,   J  =    - 


(32)  Take  the  vertex  as  origin,  the  axis  of  the  cone  as  axis  of  xr 
and  one  of  the  bounding  planes  as  plane  of  xy.  Then,  if  a  be  the 
radius  of  the  base,  h  the  height,  and  2  a  the  angle  at  the  vertex  of  the 
cone,  the  formulae  of  Art.  40  give 


,  -      0  7      -      9      sn  d>     -      o       i  —  cos 

hence         *  =    A,   .?  =    «—-Z,    s  =    «  ---  —  - 


(33)  About  2631  miles  from  the  centre. 

(34)  Regard  the  ice-cap  as  a  surface  mass  of  density  8  ;  let  Xi  be  the 
distance  of  its  centroid  from  the  earth's  centre,  ml  the  mass  of  the  ice- 
cap,  m  that  of  the  earth  alone,  and  <f>'  the  polar  distance  of  the  arctic 
circle  ;  then  the  equation  of  moments  (m  +  ml)x=  m^i  gives,  if  m\  in 
the  parenthesis  be  neglected,  x  =  (ml/m)xl=^§  sin2<£'  =  0.216  mile. 


(35)  At  the  distance  \r  from  the  centre. 


(36)  x  = 


ANSWERS. 


179 


x--   r*  +  2  r^  +  3  ^ 
~ 


4  r^-r, 

(38)  Let  Pi  be  the  volume  of  the  whole  pyramid,  F2  that  of  the  top  cut 
off,  V  that  of  the  frustum  ;  xit  x<2,  x  the  distances  of  their  centroids  from 
the  lower  base  ;  h^  h^  hz,  their  heights.    Then  the  equation  of  moments  is 
(  F,  —  yz)x  =  V&i—  y^c2.    By  geometry,  we  have  V^j  Vt=  r^/r}  ;  hence, 
(r13  —  rf)x  =  r13xi  —  r23x2j   also,  x^=  \h^  xz  =  h  +  %h2,   hl-h^  =  h, 
hi/hz  =  r1/r2.     By  means  of  these  relations,  we  find 

(~*         r*\^-k    r*~  4*1*2*  +  3^  -         ^    ^  +  2  f^g  +  3  ^ 

(n  '  T     "^^  --  '   °r  X  =  4  - 

,     .   -      ,(2a-hY 

(39)  "  =  ii' 


(40)  Let  A,  B,  C,  D  be  the  vertices  of  the  tetrahedron,  G  its 
centroid,  GI  that  of  the  face  ABC  \  let  a,  b,  c,  dt  x,  Xi  be  the  dis- 
tances of  these  points  from  the  plane  ;  and  let  the  projections  of  these 
points  on  the  plane  be  denoted  by  A',  B',  C,  £>',  G',  GJ.  Then, 
since  GGl/J)Gl  =  1/4,  and  GGl/DGl  =  (x  —  ~x^/(d—  x^,  we  have 
(x  —  x\)l(d—  Xi)=  1/4;  hence  x  =  ^($x1  +  d).  Let  E  be  the  mid- 
dle point  of  AB,  e  its  distance  from  the  plane  ;  then,  applying  a  similar 
method  to  the  triangle  ABC,wt  find  Xi  =  ^(2  e  -f  c)  =  ^(a  -f  b  +  c)  . 
Hence,  finally,  x  =  \(a  +  b  +  c  +  d). 


(41)  x  =  \h.  (43)  3'  = 

(42)  y  =  ^yi-  (44)  ^  =  f«,  ^=1^,  2  =  f^. 

(45)    (a)  x=j=±a.  (d)  x  =  ^7r'+I28a. 

9O  7T 

16       _  _      2(11:  TT  —  8) 

a,   y=a4tf.  (<?)    ^=—  ^^  --  ^^z. 

15(3^-4) 


(46)  Take  as  element  a  hemispherical  shell  of  radius  r  and  thick- 
ness dr;  x  =     **      3    a. 

(47)  x  =  \(H+h). 

(48)  Compare  problems  (40)  and  (5),  and  apply  the  propositions 
of  Pappus,  Arts.  30  and  42  ;    V—  J-TT(/  -f-  q  -f  r)A,  where  A  is  the  area 
of  the  triangle  ;  6"=  -rr\a(q-\-  r}  -f  £(r  +  /)-f-  <:(/-!-  ^)].      •../ 


!8o  ANSWERS. 

(49)  Taking  the  axis  of  the  cup  as  axis  of  x,  let  (a,  b)  be  the  cen- 
troid of  cup  and  handle,  m  their  mass ;    (xlf  o)   the   centroid  of  the 
water  whose  mass  can  be  expressed  by  (m/c)x^  where  c  is  a  constant. 
Then  the  co-ordinates  x,  y  of  the  centroid  of  cup,  handle,  and  water 
together    fulfil    the    equation    (a  -f-  c)y2  —  bxy  —  2  bey  +  b~c  =  o,    which 
represents  a  hyperbola. 

(50)  Taking  the  axis  of  z  parallel  to  the  axis  of  the  cylinder,  and  the 
origin  in  the  line  of  intersection  of  the  bases,  we  have  F==  (  \  zdxdy,  or 
if  <f>  be  the  angle  of  inclination  of  the  bases  : 

F=  tan<£  f  \ydxdy  =  tan  <f>  -y  \  \  dxdy. 

(51)  Apply  (50)  twice. 

Page  35. 

(1)  300 ooo  F.P.S.  units.  (3)   71600. 

(2)  34^  miles  per  hour. 

Page  40. 

(1)  6.4  x  io5poundals=  8.9  x  io9  dynes.  (3)  0.14. 

(2)  4.5  pounds.  (5)  60. 

Pages  51-53. 

(4)    100  V7 ;  tan-1 1  Vs. 
(7)   ioV^(V 
(9)    <2  =  i(- 


(io)  R  =  569  ;  angle  with  horizon  =  99°  27'. 

(12)  Twice  the  focal  distance. 

(13)  124°  i2f.5. 

(14)  90°. 

(15)  (a)  V~2.W;  (S)  2Wcosi(ir/2-0). 

(18)  2  Wcos  %(ir/2  +  a),  etc. 

(19)  (a)  a  =  30°,  £=120°,  7=30°;  (£)  impossible. 
(21)   T=fsW,  nearly;  ^=C=o.86W^  ^=0.675^. 


ANSWERS.  l8l 

(22)  r=iVs^;  A  =  c=tVsir;  B=W. 

(23)  S  +  3/?=T,  30°</?<6o°;  0=6o°. 

(24)  Resolve  the  components  Plt  P%  along  the  bisectors  of  0. 

(25)  J^tana;  13.4,  35.0,  107.2,  572,  oo  pounds. 

Pages  56-58. 

(2)  R  =  6,  and  acts  along  5. 

(3)  T=W-a/c,  P=W.t>/c. 

(4)  A  C  must  bisect  the  angle  BCW. 

(5)  R*  =  A2  +  B*  +  C2  +  2  BCcosa  +  2  C^  cos/?  +  2^  cosy. 

(6)  Compare  Arts.  90,  92. 

(7)  The  sum  of  their  moments  must  vanish  for  two  points  in  the 
plane  not  in  line  with  their  point  of  intersection. 

(10)  P=$W;    T=^W. 

(i  i)    T=  W;  P=  0.89  JFalong  the  bisector  of  £  BCW. 

(12)  P=  Wsm(a+/3)  sin/3  becomes  a  maximum  for  )8  =  (7r—  «)/2, 
i.e.  when  the  sail  bisects  the  angle  between  boat  and  wind. 


w 


,  (14)  Tension  in  ^^  and  CD=W'l/ 
where  ^=  V/2-  i(^-  /)  2. 

(15)  The  resultant  acts  along  the  diameter  through  A,  and  is  in 
magnitude  equal  to  the  perimeter. 

(16)  P(i  +  V2). 

(17)  (a)  WsinO,  fFcosfl;   (£)  W^tan^,  W/cosO  ;  (c) 


(20)  Produce  BO  to  the  intersection  Z>  with  the  circumscribed 
circle  ;  then  DA  is  equal  and  parallel  to  the  resultant  of  OA,  OB. 
DAO'C  is  a  parallelogram  ;  hence  DA  =  CO1. 


182  ANSWERS. 

Pages  67-68. 

(1)  Take  moments  about  the  fulcrum.     The  distance  of  this  point 
from  the  end  carrying  the  mass  12  is  (a)  3T6¥  ft. ;   (b)  3^  ft. 

(2)  (a)  A  =  i2|  tons,  £=  nj-  tons;   (V)  i8£,  i8|. 

(3)  (a)  P=  W-,   (b)  P=  (i  4-  VI)  JF. 

(6)    (<*)  19.4  tons  and  21.1  tons;   (b)  30.5,  9.9. 

(8)  Let  «  be  the  angle  subtended  at  the  centre  by  the  side  12,  and 
0  the  angle  at  which  the  diagonal  13  is  inclined  to  the  horizon;  then 

j^7  }y 

tan  6  =  ^  __  ^  esc  a  -f  cot  a. 

(9)  x  =  Fzl sin «2/(^i  sin ax  +  /^2  sin a2) . 

Page  75. 


(i)    C=i,   D  = 
=  4.2,  EF=  8.9  ;  reaction  at  A  =  4.5,  at  F=  8.9. 

(2)  H=  69.4,  T=  73.8. 

(3)  *  = 


Pages  90-92. 

(1)  T—  7.68,  ^  =  9.76,  ^=  1 2.80  pounds. 

(2)  T=  2mW,  A=  V4  ni1  —  2  m  -\-  i  W^  where  w  =  r//. 

(3)  The  three  forces   W,  T,  A  must  pass  through  a  point;  cos<£ 
=  2V|(i  -^2),  where  m  =  l/&;  T= 

(4)  r= 

(5)  Ax  =  B  = 

(6)  ^sr= 

the  thrust  ^z  in  this  case  is  to  that  in  Ex.  (5)  as  sin2a  is  to  i. 

(7)  A  =  W,   C  =  D  =  (l/a}cosOW. 

(8)  x  =  am,  A  =  V^^T  W,   C=m  W,  where  m  =  (//«)*. 


(9)  ^  =  1(30/4-  W)  tan  «,   ^=(30/4-  ^)Vitan2«+ i. 
(10)  cos0  =  \(m  +  Vw2  -f  32),  where  w  =  I/a. 


ANSWERS, 


Page  101. 


183 


(i)  The  horizontal  and  vertical  components  of  the  reaction  at  C 

are  Cx  =  --  W,   Cy=  ^^^  Wt  where  h  is  the  perpendicular  dis- 
a  -\-  b  a  -{-  b 

tance  of  C  from  AB,    and   a,   b   are    the    segments    into    which   this 
perpendicular  divides  AB. 

Pages  112-113. 

CO   60  ± 


jP<2  JFsintf;   (0  if  Pact  up  the  plane,  Jp=sm>  +       jp.  if  />  act 

cos  <£ 

down  the  plane,  />=sm(*-0)  ^ 
cos</> 

(2)   3  tons. 


cos(^-a) 

(4)  W/>=^(/-<^)  W. 

cos(«  +  ^>) 

(5)  ^  =  ^-2^. 

<•>•—  - 


Pages  145-146. 

<3)    (*)  75°°;    W   3l6°425ooo.  (5)   150  ft.-pounds. 

(4)    1  8  ooo  ft.-pounds. 

Page  164. 

(2)  The  centre  of  the  inscribed  circle. 
/  \       a  —  b 


THEORETICAL  MECHANICS 


PART  III: 
KINETICS 


PREFACE. 


ABOUT  one-half  of  this  volume  is  devoted  to  the  kinetics  of  a  particle, 
the  remainder  being  given  to  the  study  of  the  kinetics  of  a  rigid  body 
and  a  brief  discussion  of  the  fundamental  principles  of  the  kinetics  of  a 
system. 

The  first  part  of  the  chapter  on  the  motion  of  a  particle  (impact, 
rectilinear  motion)  gradually  introduces  and  illustrates  in  an  elementary 
way  such  fundamental  ideas  as  momentum,  impulse,  kinetic  energy, 
force,  work,  potential  energy,  power.  Then  the  general  equations  of 
motion  of  a  particle  are  discussed ;  and  the  principle  of  kinetic  energy 
(or  vis  viva),  that  of  angular  momentum  (or  of  areas),  and  the  prin- 
ciple of  d'Alembert  are  explained  and  applied,  first  to  the  motion  of  a 
free  particle  (central  forces),  then  to  constrained  motion.  The  example 
of  such  recent  writers  as  Budde  and  Appell  has  been  followed  in  treating 
the  constraints  of  a  particle  with  more  than  usual  fulness,  introducing 
generalized  co-ordinates,  and  establishing  the  equations  of  motion 
of  a  particle  in  the  Lagrangian  form.  It  is  believed  that  this  will 
materially  aid  the  student  in  understanding  the  use  of  these  methods 
in  the  general  case  of  the  motion  of  a  system. 

The  chapter  on  the  motion  of  a  rigid  body,  after  a  discussion  of  the 
fundamental  principles  and  of  the  theory  of  moments  and  ellipsoids 
of  inertia,  takes  up  separately  the  action  of  impulses  and  the  motion 
under  continuous  forces.  The  last  chapter,  on  the  motion  of  a  system, 
is  necessarily  brief,  owing  to  the  elementary  character  of  the  treatise. 
A  sketch  of  the  theory  of  Lagrange's  generalized  co-ordinates  and  of 
Hamilton's  principle  is,  however,  included. 

For  a  shorter  course,  the  Articles  104-159,  180-188,  190-217,  225, 
262,  268-272,  274-290,  304-310,  320-323,  327,  329-332,  336-356, 
391-397  may  be  omitted. 

ALEXANDER  ZIWET. 

ANN  ARBOR,  MICH., 
October,  1894. 


CONTENTS. 


CHAPTER  V. 
KINETICS  OF  A  PARTICLE. 

I.    IMPULSES;   IMPACT  OF  HOMOGENEOUS  SPHERES        • 
II.     RECTILINEAR  MOTION         .        .        .  ,        ;' 

III.  -FREE  CURVILINEAR  MOTION: 

1.  General  principles      •         .   .    ... 

2.  Central  forces    .         »        ..... 

3.  The  problem  of  two  bodies        . 

IV.  CONSTRAINED  MOTION: 

1.  Introduction 

2.  Motion  on  a  fixed  curve     ..... 

3.  Motion  on  a  fixed  surface  ..... 

4.  Motion  on  a  moving  or  variable  curve  or  surface 
V.    LAGRANGE'S  FORM  OF  THE  EQUATIONS  OF  MOTION  . 


PAGE 
I 

18 


38 
56 
79 

85 

87 

98 

104 


CHAPTER  VI. 

• 

KINETICS  OF  A  RIGID  BODY. 

I.    GENERAL  PRINCIPLES.        v        . 
II.     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES: 

1.  Introduction      .         .... 

2.  Ellipsoids  of  inertia  . 

3.  Distribution  of  principal  axes  in  space 


119 

132 

139 
150 


viii  CONTENTS. 

III.  RIGID  BODY  WITH  A  FIXED  Axis 

IV.  RIGID  BODY  WITH  A  FIXED  POINT: 

1.  Initial  motion  due  to  impulses   . 

2.  Continuous  motion  under  any  forces 

3.  Continuous  motion  without  forces 
V.    FREE  RIGID  BODY: 

1.  Initial  motion  due  to  impulses   . 

2.  Continuous  motion    , 


CHAPTER   VII. 
MOTION  OF  A  VARIABLE  SYSTEM. 

I.    FREE  SYSTEM 

II.    SYSTEM  SUBJECT  TO  CONDITIONS 


ANSWERS 


THEORETICAL  MECHANICS. 


CHAPTER   V. 
KINETICS   OP  A  PARTICLE. 

I.    Impulses  ;   Impact  of  Homogeneous  Spheres. 

1.  Momentum   and  Impulse.     A  particle  of   mass  m>  moving 
•with  the  velocity  v,  is  said  to  have  the  momentum  mv  (see  Part 
II.,  Art.   56).     As  long  as  this  momentum  remains  constant, 
the  particle  will  move  in  a  straight  line  with  constant  velocity 
•v  (Newton's  first  law  of  motion,  Part  II.,  Art.  74).     Any  change 
occurring  in  the  momentum  is  ascribed  to  the  action  of  a  force 
Fon  the  particle. 

2.  If  the  rate  of  change  of  momentum  is  constant  during  the 
time   t1  —  /,  the   force  F  is  constant,  and  is  measured  by  the 
•change  of  momentum  in  the  unit  of  time  ;  that  is, 

F(t'  —  t)=mvf  —  mvt  (i) 

where  v  is  the  velocity  at  the  time  /,  and  v'  the  velocity  at  the 
time  t1  (Newton's  second  law  of  motion).  As  the  product 
F(t'  —  t)  of  a  constant  force  into  the  time  during  which  it  acts 
is  called  the  impulse  of  the  force  during  this  time  (Part  II.,  Art. 
•61),  equation  (i)  can  be  expressed  in  words  by  saying  that 
the  impulse  of  the  force  is  equal  to  the  change  of  momentum. 

This  proposition  is  easily  seen  to  hold  even  for  a  variable 
force.     For  such  a  force,  we  have 


PART   III  —  I 


2  KINETICS    OF   A   PARTICLE.  [3. 

hence,  by  integration, 

Jt- 
Fdt=mv' —  mv,  (2) 

where  the  time-integral  in  the  left-hand  member  is  the  impulse 
of  the  variable  force  ^during  the  time  t'  —  t. 

3.  It  appears,  from  equations  (i)  and  (2),  that  a  very  large 
force   may   produce  a  finite   change  of  momentum  in   a  very 
short  interval  of   time,   but   that  it  would  require   an   infinite 
force   to   produce   an   instantaneous    change  of    momentum   of 
finite  amount.     The  impact  of  one  billiard  ball  on  another,  the 
blow  of  a  hammer,  the  stroke  of  the  ram  of  a  pile-driver,  the 
shock  imparted  by  a  falling  body,  by  a  projectile,  by  a  railway 
train  in  motion,  by  the  explosion  of  the  powder  in  a  gun,  are 
familiar  instances  of  large  forces  acting  for  only  a  very  short 
time  and  yet  producing  a  very  appreciable  change  of  velocity. 
The  time  of  action,  /'  —  t,  of  such  a  force  is  the  very  brief  period 
during  which  the  colliding  bodies  are  in  contact.     The  force, 
F,  is  a  pressure  or  an  elastic  stress  exerted  by  either  body  on 
the  other  during  this  time. 

Forces  of  this  kind  are  called  impulsive,  or  instantaneous, 
forces. 

4.  In  the  case  of  such  impulsive  forces,  it  is  generally  diffi- 
cult or  impossible  by  direct  observation  or  experiment  to  deter- 
mine separately  the  very  brief  time  of  action,  t1  —  t,  as  well  as 

.the  magnitude  Fvi  the  impulsive  force.  Moreover,  what  is  of 
most  practical  importance  and  interest  in  such  cases  of  impact 
is,  generally,  not  the  force  itself,  but  the  change  of  momentum 
produced,  i.e.  the  impulse  of  the  impulsive  force. 

In  the  present  section,  which  is  devoted  to  the  study  of  the 
simplest  cases  of  impact,  we  shall  therefore  deal  with  impulses 
and  momenta,  and  not  with  forces. 

5.  It  should  be  observed  that  many  authors  use  the  name  impulsive, 
or  instantaneous,  force  for  what  has  here  been  called  the  impulse  of  the 


;.]  IMPULSES.  3 

impulsive  force.     They  define  an  impulsive  force  as  the  limiting  value 

Jtr 
Fdt  when  F  increases  indefinitely,  while  at  the  same 

time  the  difference  of  the  limits,  /  —  /,  is  indefinitely  diminished  ;  in 
other  words,  as  the  impulse  of  an  infinite  force  producing  a  finite 
change  of  momentum  in  an  infinitesimal  time. 

According  to  this  definition,  an  impulsive  or  instantaneous  force  is  a 
magnitude  of  a  character  different  from  that  of  an  ordinary  force,  and  is 
measured  by  a  different  unit.  Its  dimensions  are  MLT"1,  and  not 
MLT~2.  Its  unit  is  the  same  as  the  unit  of  momentum.  Indeed,  it  is 
not  a  force,  but  an  impulse. 

We  can  arrive  at  this  idea  of  an  instantaneous  force  from  a  some- 
what different  point  of  view.  Just  as  in  kinematics  (Part  I.,  Arts.  104 
and  156)  we  may  distinguish  accelerations  of  different  orders,  regard- 
ing velocity  as  acceleration  of  order  zero,  so  in  dynamics  instantaneous 
forces  may  be  regarded  as  forces  of  order  o,  ordinary  (continuous) 
forces  as  forces  of  order  i,  the  product  of  mass  into  the  acceleration  of 
the  second  order  as  a  force  of  the  second  order,  and  so  on. 

In  the  present  elementary  treatise,  no  use  is  made  of  these  considera- 
tions. The  word  force  is  always  used  as  meaning  the  product  of  mass 

C* 

into  acceleration  of  the  first  order,  and  the  time-integral    I    Fdt  is  always 

called  impulse,  and  not  impulsive  force. 

6.  The  momentum  mv  of  a  particle  P  of  mass  m,  moving 
with  the  velocity  v,  can  be  represented  geometrically  by  a 
vector  (more  exactly  by  a  localized  vector,  or  rotor),  i.e.  by  a 
segment  of  a  straight  line  drawn  through  P  and  representing 
by  its  length  the  magnitude  of  the  momentum,  by  its  direction 
and  sense  the  direction  and  sense  of  the  velocity.  Hence,  the 
composition  and  resolution  of  momenta  follows  the  same  rules  that 
hold  for  forces. 


7.  Let  us  consider  two  particles  Plt  P^  of  masses  m^  m^ 
having  equal  and  parallel  velocities  v,  and  let  their  momenta. 
m-p,  m^v  be  represented  by  their  vectors  (Fig.  i).  The  two 
particles  may  be  regarded  as  forming  a  single  moving  system  ; 
as  the  velocities  are  equal  in  magnitude,  direction,  and  sense, 
the  system  has  a  motion  of  translation.  According  to  the  rule 


4  KINETICS   OF   A   PARTICLE.  [8. 

for   compounding  parallel   rotors    (explained   for   rotors  repre- 
senting  angular  velocities  in   Part  I.,  Arts.  253-255,  and  for 

rotors  representing  forces  in  Part  II., 
Arts.  104-107),  the  resultant  momen- 
tum is  parallel  to  the  given  momenta 
and  equal  in  magnitude  to  their  alge- 
braic sum  (m1  -f-  m%)  v  ;  and  its  line 
divides  the  distance  P\P%  in  the 
inverse  ratio  of  the  momenta  m^ut 
m<p,  or  of  the  masses  mlt  m2.  The 
resultant  passes,  therefore,  through  the 
centroid  P  of  the  masses  m^  m2. 

8.  It  is  easy  to  see  how  this  proposition  can  be  generalized. 
If  any  number  of  particles,  all  having  equal  and  parallel  veloci- 
ties, be  given,  the  resultant  momentum,  or  the  momentum  of  the 
system,  is  equal  to  the  mass  of  the  system  multiplied  by  the 
common   velocity,   and   passes   through    the    centroid    of    the 
system. 

Thus,  in  the  case  of  a  rigid  body  having  a  velocity  of  transla- 
tion v,  but  no  rotation,  the  whole  mass  M  of  the  body  may  be 
regarded  as  concentrated  at  the  centroid,  and  the  momentum  of 
the  centroid,  Mvt  is  then  equal  to  that  of  the  body. 

9.  But  we  can  speak  of  the  momentum  of  a  system  of  par- 
ticles even  when  their  velocities  are  not  of  equal  magnitude 
but  only  parallel. 

Let  x  be  the  distance,  at  the  time  /,  of  any  particle  P  of  mass 
m  from  some  fixed  plane,  which,  for  the  sake  of  simplicity,  we 
may  take  at  right  angles  to  the  direction  of  the  velocity.  Then 
the  distance  x  of  the  centroid  G  of  the  system  at  the  time  / 
from  the  same  plane  is  (Part  II.,  Art.  13) 


-  _.       _  /  x 

—    •C1          —       Ti/r    '  *•" 

2,m        M 

Differentiating  this  equation  with  respect  to  the  time,  and  re- 


ii.]  IMPACT   OF   SPHERES.  5 

membering  that  dx/dt=v  is  the  velocity  of  the  particle  P,  we 
find  for  the  velocity  dx/dt=v  of  the  centroid 


10.  In  the  special  case  of  two  particles  /\,  P%  of  masses  mlt 
mz,  moving  with  the  velocities  vlt  v%  in  the  same  straight  line, 
we  have 

7>  =  mivi  +  m*v*_  (  } 


If  the  velocities  vlt  v^  be  constant,  this  equation  shows  that  the 
centroid  moves  with  constant  velocity  and  constant  momentum 
in  the  same  line. 

Similarly,  the  more  general   equation  (4)  of  the  preceding 
article, 

Mv  =  *£mv,  (6) 

shows  that  the  momentum  of  a  system  of  particles  moving  with 
constant  velocities  in  the  same  direction  remains  constant,  i.e.  the 
centroid  of  such  a  system  moves  with  constant  velocity  in  a 
straight  line.  It  is  to  be  noticed  that  the  velocities  need  not  be 
all  of  the  same  sense  ;  that  is,  v  may  be  positive  for  some  par- 
ticles and  negative  for  others. 

This   proposition   may  be   regarded   as   a   generalization  of 
Newton's  first  law  of  motion. 

11.  Direct  Impact.  We  proceed  to  consider  the  particular 
case  of  two  homogeneous  spheres  of  masses  m,  m't  whose  centres 
C,  C'  move  with  velocities 
u,  it1  in  the  same  straight 
line.  The  spheres  are  sup- 
posed not  to  rotate  but  to 
have  a  motion  of  pure  trans- 
lation ;  then  their  momenta 
are  mu,  m!u'  >  and  can  be  £' 

represented  by  two  vectors  drawn  from  the  centres  C,  C  along 
the  line  CO  (Fig.  2).  To  fix  the  ideas'  we  assume  the  velocities 


6  KINETICS    OF   A   PARTICLE.  [12. 

u,  u*  to  have  the  same  sense  and  u>u',  so  that  m  will  finally 
impinge  upon  m'.  The  case  when  the  velocities  are  of  oppo- 
site sense  will  not  require  special  investigation,  as  only  the 
sign  of  u'  would  have  to  be  changed. 

//  is  our  object  to  determine  the  velocities  v,  v'  of  m,  m',  im- 
mediately after  impact,  when  the  velocities  _«,  «'  immediately 
before  impact  are  given. 

The  results  here  derived  for  homogeneous  spheres  hold, 
generally,  whatever  the  shape  of  the  impinging  bodies,  provided 
that  they  do  not  rotate,  and  that  the  common  normal  at  the 
point  of  contact  passes  through  both  centroids. 

12.  If  the  spheres  were  perfectly  rigid,  the  problem  would  be 
indeterminate,  for  there  is  no  way  of  deciding  how  the  velocities 
would  be  affected  by  the  collision. 

Natural  bodies  are  not  perfectly  rigid.  The  effect  of  the 
impact  will,  in  general,  consist  in  a  compression  of  the  portions 
of  the  bodies  brought  into  contact.  Moreover,  all  natural  bodies 
possess  a  certain  degree  of  elasticity  ;  the  compression  will 
therefore  be  followed  by  an  extension,  each  sphere  tending  to 
regain  its  shape  at  least  partially. 

The  compression  acts  as  a  retarding  force  on  the  impinging 
sphere  m,  and  as  an  accelerating  force  on  m'  .  It  will  last 
until,  the  velocities  #,  u1  have  become  equal,  say  =w.  During 
the  subsequent  period  of  extension,  or  restitution,  the  elastic 
stress  still  further  diminishes  the  velocity  of  my  and  increases 
that  of  m',  until  they  become,  say,  v,  v'. 

13.  The  stress  varies,  of  course,  during  the  whole  time  r  of 
compression  and  restitution.     But,  according  to  Newton's  third 
law,  the  pressure  F  exerted  at  any  instant  by  m  on  m'  must  be 
equal  and  opposite  to  the  pressure  F'  exerted  by  m'  on  m  at 
the  same  instant.     Since  F=  mdu/dt,  F1  =  m'du'/dt,  and  F=  -  F1 
at  any  instant  during  the  time  r,  we  have 


,  or  m  frdu  =  -m'  Cdu', 

t/o  JQ 


1  6.]  IMPACT   OF    SPHERES.  7 

whence  mv  —  mu=—(m'v'  —  m'u'), 

or  mv  +  m'v1  =  mti+m'u*  ;  (7) 

that  is,  the  total  momentum  after  impact  is  equal  to  that  before 
impact. 

14.  This  proposition  will  evidently  hold  for  any  number  of 
spheres  whose  centres  move  in  the  same  line,  and  can  then 
be  expressed  in  the  form 

^mv  —  ^mu.  (8) 

It  can  be  regarded  as  a  special  case  of  the  so-called  principle 
of  the  conservation  of  the  motion  of  the  centroid  to  be  proved 
hereafter  for  any  system  not  acted  upon  by  external  forces. 
On  the  other  hand,  the  proposition  can  be  looked  upon  as  a 
further  generalization  of  Newton's  first  law  of  motion.  While 
the  latter  asserts  that  the  momentum  of  a  particle  remains 
unchanged  as  long  as  no  external  forces  act  upon  it,  our  law  of 
impact  asserts  the  same  thing  for  the  momentum  of  a  system. 

15.  If  the  spheres  were  perfectly  non-elastic,  there  would  be 
only  compression  and  no  subsequent  extension.     As  at  the  end 
of   the   period   of   compression,  the  velocities  u,  u'  have  both 
become   equal,    viz.   =w   (Art.   12),  the   spheres   after   impact 
would  have  the  common  velocity 

/  x 
(9) 


m  +  m' 

.  / 

16.  If  the  spheres  were  perfectly  elastic,  i.e.  if  the  elastic  stress 
following  the  compression,  or  the  so-called  force  of  restitution, 
were  just  equal  to  the  preceding  stress  of  compression,  the 
spheres  would  completely  regain  their  original  shape.  In  this 
case,  the  elastic  stress  causes  the  impinging  sphere  m  to  lose 
during  the  period  of  restitution  an  amount  of  momentum 
m(w  —  u}  equal  to  that  lost  during  the  period  of  compression. 
Hence,  the  final  velocity  of  m  after  impact  would  be 


8  KINETICS   OF  A   PARTICLE.  [17. 

Similarly,  we  have  for  the  other  sphere  m* 

v'  =  w+(w  —  u')  =  2  w  —  ti'. 

As  w  is  known  from  (9),  the  velocities  after  impact  can  be 
determined  by  means  of  these  formulae  for  perfectly  elastic 
spheres. 

17.  In  general,  physical  bodies  are  imperfectly  elastic,  the  force 
of  restitution  being  less  than  that  of  the  original  compression ; 

that  is,  we  have 

(w — v)  =  e(u  —  w), 

(v'  —  w)=  e(w  —  ur), 

where  e  is  a  proper  fraction  whose  limiting  values  are  o  for 
perfectly  inelastic  bodies  and  i  for  perfectly  elastic  bodies. 
This  fraction  e,  whose  value  for  different  materials  must  be 
determined  experimentally,  is  called  the  coefficient  of  restitution 
(or  less  properly,  the  coefficient  of  elasticity). 

18.  To  eliminate  w  we  have  only  to  add  the  last  two  equa- 
tions ;  this  gives 

v1  —  v  =  e(u  —  u')'y  (10) 

that  is,  the  ratio  of  the  relative  velocity  after  impact  to  the  rela- 
tive velocity  before  impact  is  constant  and  equal  to  the  coefficient 
of  restitution. 

This  proposition,  in  connection  with  the  proposition  of  Art. 
13,  expressed  by  formula  (7),  is  sufficient  to  solve  all  problems 
of  so-called  direct  impact,  i.e.  when  the  centres  of  the  spheres 
move  in  the  same  line. 

19.  As  the  coefficient  e  is  frequently  difficult  to  determine, 
the  limiting  cases  e=o,  e=i  are  important  as  giving  approxi- 
mate solutions  for  certain  classes  of  substances. 

Thus,  for  nearly  inelastic  bodies  (such  as  clay,  lead,  etc.) 
we  may  put  e=o,  whence,  by  (10),  v1  =  v,  i.e.  the  velocities  of 
the  spheres  become  equal  after  impact ;  and  the  value  of  the 
common  velocity  is  found  from  (7)  as 

mu-\-m'uf 

v= 


20.]  IMPACT   OF   SPHERES.  9 

which  agrees  with  the  result  (9)  found  in  Art.  15.  For  per: 
fectly  elastic  bodies  e=i,  and  formula  (10)  shows  that  in  this 
case  the  relative  velocity  after  impact  is  numerically  equal  to 
that  before  impact,  but  of  opposite  sense. 

20.   Exercises. 

(1)  Two  balls  of  clay  (<?=o)  weighing  2  and  3  oz.  move  in  the  same 
direction.     The  heavier  ball  impinges  from  behind  upon  the  lighter  ball 
at  the  moment  when  the  latter  moves  at  the  rate  of  15  ft.  per  second. 
If  the  velocity  of  the  lighter  ball  is  doubled  by  the  impact,  what  was  the 
original  velocity  of  the  heavier  ball  ? 

(2)  Two  glass  balls  (<?=i)  weighing  i   Ib.  and  12  oz.,  respectively, 
move  in  the  same  line  with  velocities  of  5  and  4  ft.  per  second.     What 
are  their  velocities  after  impact  (a)  if  their  original  velocities  were  of 
the  same  sense,  (<£)  if  they  were  of  opposite  sense  ? 

(3)  A  ball  weighing  5  Ibs.,'  while  moving  with  a  speed  of  51  ft.  per 
second,  overtakes  a  ball  of  7  Ibs.  moving  in  the  same  line  at  the  rate  of 
40  ft.  per  second.      If  the  coefficient  of  restitution  be  ^,  what  are  the 
velocities  of  the  two  balls  after  impact  ? 

(4)  With  the  data  of  Ex.  (3),  show  that  the  velocities  after  impact 
would  be  equal  if  the  balls  were  perfectly  inelastic,  and  that  these  veloci- 
ties would  differ  more  than  in  Ex.  (3)  if  the  balls  were  perfectly  elastic. 

(5)  Find  the  velocity  with  which  an  elastic  ball  rebounds  from  a 
fixed  surface  after  impinging  upon  it  perpendicularly  with  a  velocity  u. 

(6)  To  determine  the  coefficient  of  restitution,  a  ball  is  dropped 
from  a  height  H  on  a  fixed  horizontal  plate  of  the  same  material,  and 
the  height  of  rebound  h  is  measured.     Show  that  e  =  ^Jh/H. 

(7)  A  ball  is  dropped  from  a  height  H~  12  ft.  on  a  fixed  horizontal 
plate.     Find  the  height  h  to  which  it  will  rebound  if  e  =  f . 

(8)  If  not  disturbed,  the  ball  in  Ex.  (7)  will  continue  to  fall  and 
rebound  alternately,     (a)  What  height  does  it  reach  at  the  tenth  re- 
bound?    (3)   In  what  time  does  it  come  to  rest?     (c)   What  is  the 
whole  space  described? 

(9)  A  number  of  equal,  perfectly  elastic  balls  are  placed  in  contact  so 
that  their  centres  are  in  a  straight  line.     An  equal  ball  impinges  with  a 
velocity  u  along  this  line  on  the  first  ball  of  the  row.     Show  that  the 


10  KINETICS    OF   A  PARTICLE.  [21. 

last  ball  of  the  row  will  move  off  with  the  velocity  u,  while  all  the  other 
balls  will  remain  at  rest. 

(10)  Find  the  velocity  of  the  last  («th)  ball  in  Ex.  (9),  when  the 
coefficient  of  restitution  is  e. 

(n)  An  inelastic  ball  of  8  Ibs.  is  moving  with  a  velocity  of  12  ft.  per 
second,  (a)  With  what  velocity  must  a  ball  of  24  Ibs.  meet  it  to  arrest 
its  motion?  (£)  With  what  velocity  would  the  ball  of  24  Ibs.  have  to 
impinge  from  behind  on  the  ball  of  8  Ibs.  to  double  its  velocity  ? 

(12)  A  ball  m  impinges  upon  a  ball  m'  from  behind  with  a  velocity 
u.  Determine  the  velocities  after  impact,  both  for  inelastic  and  for  per- 
fectly elastic  balls  :  (a)  when -m'  is  originally  at  rest;  (b)  when  m'  is 
at  rest  and  very  large  in  comparison  with  m ;  (<r)  when  m1  has  the 
initial  velocity  z/',  and  is  very  large  in  comparison  with  m. 

21.  Kinetic  Energy.     A  particle  of  mass  mt  moving  with  the 
velocity  v,  has  the  kinetic  energy  ^mv2  (Part  II.,  Art.  71).     As 
this  is  not  a  vector-quantity,  the  kinetic  energy  of  a  system 
consisting  of  any  number  of  free  particles  is  simply  the  alge- 
braic sum,  ^^mv^y  of  the  kinetic  energies  of  these  particles. 
It  is  an  essentially  positive  quantity,  provided  the  masses  are 
all  positive. 

The  kinetic  energy  of  a  rigid  body  having  a  motion  of  pure 
translation  is  evidently  =\mviy  if  m  be  the  mass  of  the  body 
and  v  the  common  velocity  of  all  its  points. 

22.  Change  of  kinetic  energy  is  brought  about  by  the  action 
of  force,  and  we  have  (Part  II.,  Art.  72)  for  a  constant  force  Ft 

\mv'*-\mv*=F(s>-s);  (u) 

and  for  a  variable  force  F, 

'  (12) 

where  the  quantity  in  the  right-hand  member  is  called  the  work 
of  the  force.  Thus  a  particle,  of  mass  m,  falling  from  rest 
through  a  distance  s,  acquires  its  kinetic  energy  owing  to  the 


IMPACT   OF   SPHERES.  U 

work  done  upon  it  by  the  constant  attractive  force,  F—mg^  of 
the  earth,  and  we  have 

l  mv2 —  Fs  =  mgs. 

The  kinetic  energy  \  miP,  possessed  by  a  particle  of  mass  m, 
moving  with  the  velocity  v,  can  therefore  always  be  regarded  as 
equivalent  to  a  certain  amount  of  work.  If  the  motion  of  this 
particle  be  opposed  by  a  constant  force  or  resistance  F,  the  dis- 
tance s  through  which  it  will  go  on  moving  until  it  comes  to 
rest  is  of  course  determined  from  the  same  equation, 

±mv*=Fs.  (13) 

It  is  then  said  that  the  kinetic  energy  of  the  particle  is  spent  in 
overcoming  the  resistance  F,  or  in  doing  work  against  the  force 
^(see  Part  II.,  Art.  231). 

23.  In  the  case  of  direct  impact  of  spheres,  as  considered  in 
Art.  11,  the  velocity,  and  hence  also  the  kinetic  energy,  of  each 
sphere  is  in  general  changed  by  the  impact ;  a  transfer  of  kinetic 
energy  can  be  said  to  take  place.     Thus,  when  a  sphere  at  rest 
is  struck  by  a  moving  sphere,  kinetic  energy  is  imparted  to  the 
former  by  the  impulsive  force,  and  this  energy  can  then  be 
spent  in  doing  work  against  a  resistance.     Impact  is  therefore 
frequently  used  for  the  purpose  of  performing  useful  work. 

24.  For  instance,  to  drive  a  nail  into  a  wooden  plank,  the 
resistance  F  of  the  wood  must  be  overcome  through  a  certain 
distance  s.     This  might  be  done  by  applying  a  pressure  equal 
to  F ;  as,  however,  this  pressure  would  have  to  be  very  large,  it 
is  more  convenient  to  impart  to  the  nail,  by  striking  it  with  a 
hammer,  an  amount  of  kinetic  energy,  ^mv2,  equivalent  to  the 
work  Fs  that  is  to  be  done.     Neglecting  elasticity,  and  denoting 
the  mass  of  the  hammer  by  m,  that  of  the  nail  by  mf,  the  veloc- 
ity of  the  hammer  at  the  moment  when  it  strikes  the  head  of 
the  nail  by  «,  we  have,  by  (7), 

mv  +  m'v'  =  mu, 


! 


12  KINETICS   OF   A  PARTICLE.  [25. 

or  since,  by  (10),  for  inelastic  impact  v'  =  v, 


m 


. 

m+m* 

This   is   the   common  velocity  of   hammer   and  nail  after  the 
stroke.     We  find,  therefore,  by  (13), 


mil 


-^-f.^^  =  Fs.  (14) 

m+m1 

25.  It  will  be  noticed  that  while  the  total  kinetic  energy 
of  hammer  and  nail  just  before  striking  was  ^mu2+o,  the 
kinetic  energy  utilized  for  driving  the  nail  is  only  the  fraction 
m/(m+mf)  of  this  total  kinetic  energy.  The  remaining  portion 
of  the  original  kinetic  energy,  viz. 

(15) 


m  +  m1 

must  be  regarded  as  spent  in  producing  the  slight  deformations 
of  hammer  and  nail  and  such  accompanying  phenomena  as 
vibrations  of  the  plank,  sound,  heat,  etc.  For  it  is  an  experi- 
mental result  of  modern  physical  research  that,  wherever  kinetic 
energy  disappears  as  such,  there  is  done  an  exactly  equivalent 
amount  of  work.  The  apparently  disappearing  kinetic  energy 
may  either  be  transferred  to  some  other  body,  as  in  the  case  of 
the  vibrations  of  the  plank,  or  it  may  reappear  in  the  form  of 
molecular  vibrations,  causing  sound  or  heat ;  or  it  may  be  trans- 
formed into  an  equivalent  amount  of  so-called  potential  energy. 
This  physical  fact  is  known  as  the  principle  of  the  conservation 
of  energy. 

26.  In  our  example  the  total  original  kinetic  energy,  Ji»«*> 
resolves  itself  into  two  portions,  the  portion  (14)  used  for  driv- 
ing the  nail,  and  the  "wasted"  or,  as  it  is  often  called,  "lost" 
portion  (15).  It  may,  however,  happen  that  the  portion  (15) 


28.]  IMPACT   OF   SPHERES.  !3 

does  the  useful  work,  while  (14)  is  wasteful.  This  would  be  the 
case,  for  instance,  in  molding  a  rivet  with  a  hammer,  or  in  forg- 
ing a  piece  of  iron  under  the  blows  of  a  steam-hammer.  The 
useful  work  here  consists  in  the  deformation  of  the  bolt  or  piece 
of  iron. 

It  appears  from  the  expressions  (14)  and  (15)  that,  for  the 
purpose  of  driving  the  nail,  m  should  be  large  in  comparison 
with  m',  while  for  molding  a  rivet  it  is  of  advantage  to  have  m' 
large  in  comparison  with  m. 

27.  In  applying  the  formulae  (11)  to  (15),  and  in  general  all 
formulae  of   theoretical  kinetics,  it  should  be  noticed  that  the 
forces  are  supposed  to  be  expressed  in  absolute  measure,  the 
unit  being  the  poundal  or  the  dyne.    Hence,  to  find  the  force  F 
in  pounds  the  numerical  result  obtained  from    one   of   these 
formulae  must  be  divided  by  the  value  of  g. 

28.  Let  us  now  consider  the  change  of  the  total  kinetic  energy 
produced   by  direct   impact    in   two   partially   elastic   spheres. 
With  the  notations  of  Art.  n,  we  have  for  the  excess  of  the 
kinetic  energy  after  impact  over  that  before  impact  : 


To  eliminate  v  and  v'  from  this  expression,  square  the  equations 
</)  and  (10), 


multiply  the  latter  by  mm',  and  write  it  in  the  form 

mm'(p-v'y+(\-^mm'(p-u')* 
finally  add  the  former  equation, 


whence 

(«-«')a.    (16) 


I4  KINETICS   OF   A   PARTICLE.  [29. 

As  the  right-hand  member  of  this  equation  is  essentially 
negative,  it  appears  that  while  in  impact  the  total  momentum 
remains  unchanged,  the  total  kinetic  energy  is  in  general  dim- 
inished; only  in  the  limiting  case  of  perfectly  elastic  bodies 
(e=i)  does  the  kinetic  energy  remain  the  same  after  as  before 
impact.  The  "lost"  kinetic  energy  (16)  mainly  represents 
the  amount  of  energy  spent  in  producing  the  permanent  defor- 
mation of  the  impinging  bodies. 

29.  Exercises, 

(1)  A  hammer  weighing  1.5  Ibs.  strikes  a  nail  weighing  ^  oz.  with 
a   velocity  of  20  ft.  per  second,  and  drives  it  J  in.     Find  the  mean 
resistance  of  the  wood,  and  determine  the  useful  and  wasteful  work. 

(2)  In  Art.  20,  Ex.  (3),  find  the  loss  of  kinetic  energy  due  to  the 
impact. 

(3)  A  train  of  120  tons  runs,  with  a  speed  of  15  miles  an  hour,  into  a 
train  of  80  tons  at  rest.     Neglecting  elasticity,  determine  the  destructive 
work  of  the  collision,  and  the  velocity  along  the  track  after  impact. 

(4)  A  pile  weighing  m'  Ibs.  is  driven  into  the  ground  by  a  ram  of  m 
Ibs.,  falling  from  a  height  of  h  ft.     If  the  pile  sinks  s  in.  into  the  ground 
after  n  falls  of  the  ram,  show  that  the  resistance  of  the  ground  (assumed 

as  uniform)  is  =  l?-^[ — *      ,  ,     )  m'h  pounds. 
s    \i  +  m'/mj 

(5)  If,  in  Ex.  (4),  the  elasticity  of  ram  and  pile  be  neglected,  ram 
and  pile  will  have   equal  velocities  after  impact,  and  move  together. 
Hence,  the  factor  m'  should  be  replaced  by  m  +  m1,  and  the  resistance  is 

-  U1L J mh  pounds. 

s       i  +  m'/m 

(6)  TO  blows  of  a  ram  of  500  Ibs.,  falling  from  a  height  of  5  ft.,  sink 
a  pile  of  400  Ibs.  4  in'.      If  the  permanent  load  of  a  pile  be  taken  as 
one-fifth  of  the  resistance,  what  permanent  load  can  the  pile  bear  ? 

(7)  A  steam-hammer  of  3  tons  is  used  in  forging.     It  has  a  fall  of 
5  ft.     If  the  weight  of  the  anvil  be  20  tons,  what  is  the  useful  and  what 
the  wasteful  work? 

30.  Recoil.    The  explosion  of  the  powder  in  a  gun  produces 
an  impulsive  pressure  both  on  the  shot  and  on  the  body  of  the 


3i.]  IMPACT   OF   SPHERES.  15 

gun.  Assuming  the  line  of  motion  of  the  centroid  of  the  shot 
to  pass  through  the  centroid  of  the  gun,  we  may  apply  equation 
(7),  with  u  =  o,  u'  =  o.  Hence,  denoting  by  m  the  mass  of  the 
gun,  by  m'  that  of  the  shot,  we  find  for  the  velocity  of  recoil 


(17) 


The  kinetic  energies  ^  mv*  and  |-  m'v'^  of  gun  and  shot  are  in 
the  ratio  m1  /m  ;  hence,  the  energy  of  recoil  is  the  fraction 
m'/(m  +  mf)  of  the  total  energy  J  mv*  +  ^  m'v'2  of 'the  explosion 
of  the  powder,  while  the  energy  of  the  shot  is  =m/(m-}-m!)  of 
the  total  energy.  In  large  guns  the  recoil  is  diminished  by  a 
special  elastic  cushion  or  "compressor."  Moreover,  the  mass 
of  the  powder  gases  cannot  be  entirely  neglected  in  all  cases. 

31.  Oblique  Impact.  In  ihe  case  of  oblique  impact,  i.e.  when  the 
centres  of  the  colliding  spheres  do  not  move  in  the  same  straight 
line,  the  velocities  after  impact  can  be  found  without  difficulty, 
provided  that  the  velocities  of  the  centres  before  impact  lie  in 
the  same  plane  and  that  the  spheres  are  perfectly  smooth. 

With  these  assumptions,  let  m,  m'  be  the  masses  of  the  two 
spheres ;  C,  C'  their  centres  (Fig.  3) ;  ut  u'  the  velocities  before 
impact ;  a,  af  the  angles 
made  by  ?/,  u'  with  the  line 
CC' ;  v,  v'  the  velocities 
after  impact ;  and  0,  0' 
the  angles  they  make 
with  CC. 

As  there  is  no  friction, 

the  forces   of   impact  act  Fig.  3? 

along   the   line    CC   that 

joins  the  centres.  Hence,  resolving  each  velocity  along  and 
perpendicular  to  CO,  the  components  at  right  angles  to  CO 
must  remain  unchanged  by  the  collision  ;  that  is,  we  must  have 

v  sin/3=&  sin  a,  v'  sin£'  =  «'  sin  a',  (18) 


16  KINETICS   OF   A   PARTICLE.  [32. 

The  components  of  the  velocities  along  CO  must  satisfy  the 
equations  (7)  and  (10).  Hence,  substituting  u  cos  a,  «'  cos  a', 
v  cos  ft  z/  cos/3'  for  z/,  V,  z/,  z/,  respectively,  we  must  have 

mv  cos  ft  +  m'vf  cos  @'  =  mu  cos  a  +  m'ti'  cos  «',          (19) 
z/  cos  /3'  —  v  cos  ^  —  e(u  cos  a  —  uf  cos  a').  (20) 

32.  The  particular  case  of  the  oblique  impact  of  a  homoge- 
neous  sphere  against   a  smooth  fixed   plane  deserves   special 
mention.     In  this  case,  «'  and  v'  are  zero  ;  and  the  angles  a,  /3 
made  by  the  velocities  u,  v  with  the  normal  to  the  plane,  are 
called  the  angle  of  incidence  and  of  reflection,  respectively. 

The  equations  (18)  and  (20)  reduce  to  the  following: 

v  sin  j3  =  u  sin  a,  v  cos  /3=  —  eu  cos  a,  (21) 

where  the  minus  sign  indicates  that  the  projections  of  u  and  v 
on  the  normal  have  opposite  sense.  Dividing  the  former  of 
these  equations  by  the  latter,  we  find 

tan  a=  —  e  tan  ft  (22) 

where  the  minus  sign  merely  indicates  that  the  angles  a,  /3  lie 
on  opposite  sides  of  the  normal. 

For  perfectly  elastic  bodies,  the  last  equation  shows  that  the 
angles  of  incidence  and  reflection  are  equal. 

33.  Exercises. 

(1)  A  baseball  weighing  5^  oz.,  while  moving  with  a  velocity  of  100 
ft.  per  second  is  struck  by  the  bat  in  a  direction  at  right  angles  to  its 
line  of  motion.     Find  the  momentum  imparted  by  the  blow  if  it  deflects 
the  ball  through  an  angle  of  60°. 

(2)  Determine  the  velocity  of  recoil  of  a  gun  weighing  1500  Ibs.  when 
a  i2-lb.  shot  is  fired  from  it  with  an  initial  velocity  of  2000  ft.  per 
second. 

(3)  The  heavier  one  of  two  ivory  balls  (e  =  0.88),  whose  centroids 
are  C,  O  and  whose  masses  are  i   Ib.  and  j  lb.,  impinges  upon  the 
lighter.     The  velocity  of  the  heavier  ball  is  15  ft.  per  second  and  makes 
an  angle  of  30°  with  the  line  CO,  while  the  velocity  of  the  lighter  ball  is 
5  ft.  per  second  and  makes  an  angle  of  60°  with  the  line  CO  (pro- 
duced).    Find  the  velocities  after  impact  in  magnitude  and  direction. 


34-]  IMPACT   OF   SPHERES.  !7 

34.  As  a  more  careful  study  of  the  theory  of  impact  requires 
some  knowledge  of  the  theory  of  elasticity,  it  is  generally 
treated  more  at  length  in  works  on  applied  mechanics.  See, 
for  instance,  J.  WEISBACH,  Mechanics  of  engineering,  translated 
by  E.  B.  Coxe,  New  York,  Van  Nostrand,  1875,  Vol.  I.,  pp. 
667-711  ;  A.  RITTER,  Technische  Mechanik,  Leipzig,  Baum- 
gartner,  1884,  pp.  585-618;  J.  H.  COTTERILL,  Applied  Me- 
chanics, London,  Macmillan,  1884,  pp.  274-280  and  374-386; 
THOMSON  and  TAIT,  Natural  philosophy,  I.,  Part  I,  pp.  274-284. 
The  general  theory  of  impulsive  forces  will  be  given  in  Chapter 
VI. 


PART  III— 2 


18  KINETICS   OF   A   PARTICLE.  [35, 


II.    Rectilinear  Motion. 

35.  The  motion  of  a  single  particle  presents  a  comparatively 
simple  problem,  because  the  forces,  being  in  this  case  all  applied 
at  one  and  the  same  point,  have  a  single  resultant   which  is 
readily  found  by  geometrically  adding  the  forces  (Part  II.,  Art. 
96).     Let  this  resultant  be  denoted  by  F,  the  mass  of  the  par- 
ticle  by  m,  and  its  acceleration  by  j ;   then,  according  to  the 
definition  of  force  (Part  II.,  Art.  60),  we  must  have 

mj—F. 

This  equation  merely  expresses  the  fact  that  the  force  F  pro- 
duces in  the  mass  m  an  acceleration  /,  which  agrees  with  F  in 
direction  and  sense,  and  is  inversely  proportional  to  m. 

36.  The  forces,  whose  resultant  is  F,  are  usually  called  the 
impressed  forces.     Both  F  and/  are,  in  general,  variable.     If  at 
any  time  t  the  particle  m  were  acted  upon  by  a  force  =  —mj, 
in  addition  to  the  impressed  forces,  it  would  evidently  be  in 
equilibrium.     The  product  mj  of  the  mass  of  the  particle  into 
its  acceleration  at  any  instant  is  called  the  effective  force  of  the 
particle  at  this  instant.     It  can,  therefore,  be  said  that  the  im- 
pressed forces  are  at  any  instant  in  equilibrium  with  the  effective 
force  reversed. 

This  obvious  proposition  forms  the  fundamental  idea  of  a 
most  important  method  of  treating  the  dynamical  equations  of 
motion,  known  as  d'Alembert's  principle,  which  will  be  discussed 
more  fully  later  on  (see  Arts.  97-103,  383-386).  It  makes  it 
possible  to  apply  to  kinetic  problems  the  statical  conditions  of 
equilibrium.  Thus,  in  the  case  of  a  single  particle,  if  the  re- 
versed effective  force,  —mj,  be  combined  with  the  impressed 
forces,  we  have  a  system  of  forces  acting  on  the  particle  which, 
at  the  instant  considered,  is  in  equilibrium,  and  must  satisfy 
the  conditions  of  equilibrium  for  concurrent  forces  (Part  II., 
Arts.  97,  101),  viz.  —  mf+F=o;  or,  resolving/  into  its  com- 


38.]  RECTILINEAR   MOTION.  X9 

ponents  dzx/dfi,  d^y/dt2,  d2z/dfi,  and  .Finto  the  components  X, 

'     ' 


37.  To  familiarize  the  student  with  the  idea  of  force  and  its 
use  in  kinetics,  we  shall  now  study  in  some  detail  the  simple 
case  of  rectilinear  motion.     The  next  section  will  be  devoted  to 
the  general  problem  of  the  curvilinear  motion  of  a.  free  particle. 
This  will  be  illustrated  by  the  important  case  of  motion  due 
to  central  forces.     Finally,  the  motion  of  a  particle  subject  to 
conditions,  or  constraints,  will  be  treated. 

38.  When  a  particle  of  mass  m  moves  in  a  straight  line,  both 
its  velocity  v  and  the  resultant  force  F  must  be  directed  along 
this  line.     The  acceleration  in  rectilinear  motion  (see  Part  I., 
Art.   103)  is  j  =  dv  /  dt  =  d*s  /  dt*  ',   hence  the  dynamical  equation 
of  rectilinear  motiony 

-%-««-%-*        ^       «> 

It  differs  from  the  kinematical  equation  (Part  I.,  Art.  1  1  5)  only 
by  the  factor  m,  and  can  be  treated  in  the  same  way. 

Thus,  if  the  law  of  force  be  given,  i.e.  if  F  be  known  as  a 
function  of  t,  s,  v,  or  of  only  one  or  two  of  these  quantities,  the 
equation  can  be  integrated  ;  and  if,  moreover,  the  initial  position 
and  velocity  of  the  particle  be  given,  the  constants  of  integra- 
tion can  be  determined,  and  all  the  circumstances  of  the  motion 
can  be  found. 

If  the  mass  m  of  the  moving  particle  were  not  a  constant 
quantity,  the  equation  (i)  should  be  written  in  the  form 

d(mv)  _  p 

~dT    *' 

since  the  resultant  force  is  the  rate  at  which  the  momentum  of 
the  particle  changes  with  the  time  (see  Part  II.,  Art.  60). 


20 


KINETICS   OF   A   PARTICLE. 


[39- 


39.  As  long  as  a  single  free  particle  only  is  considered,  there 
is  generally  no  advantage  in  introducing  the  idea  of  force  ;  the 
equation  of  motion  can  be  divided  by  m,  and  this  reduces  it  to 
a  purely  kinematical  form. 

Thus,  for  a  particle  of  mass  m  falling  in  vacuo,  the  dynamical 
equation  of  motion  is 


where  W=-mg  is  the  weight  of  the  particle;  i.e.  the  force  of 
attraction  exerted  by  the  earth  on  the  particle  (in  poundals  or 
dynes,  if  m  be  expressed  in  pounds  or  grammes,  see  Part  II., 
Art.  115).  Dividing  by  m,  we  find  the  kinematical  equation 


which  has  been  treated  in  Part  I.,  Arts.  107-114. 

The  following  articles  give  examples  in  which  it  is  more  con- 
venient to  retain  the  idea  of  force. 

40.  Let  us  consider  a  mass  m  that  is  being  raised  or  lowered 
by  means  of  a  rope  or  chain  (Fig.  4),  such  as  a  building  stone 
suspended  from  a  derrick.  The  rope  acts  as  a 
constraint)  conditioning  the  motion  of  the  stone. 
To  make  the  stone  free  we  may  imagine  the 
rope  cut  just  above  the  stone  and  the  tension 
of  the  rope,  T,  introduced  as  a  substitute.  The 
stone  then  moves  under  the  action  of  two  forces, 
viz.  its  weight  W=mg  and  the  tension  T  of 
the  rope.  Taking  the  downward  sense  as  posi- 
tive, we  have  the  equation  of  motion, 


Fig.  4. 


(2) 


41.    Writing/ for  the  acceleration  d*s/dtz  with  which  the  stone 
is  being  lowered  or  raised,  we  find  for  the  tension  T  of  the  rope 

T=m(g-j).  (3) 


43-]  RECTILINEAR   MOTION.  2I 

This  equation  shows  that  the  tension  is  equal  to  the  weight  of 
the  stone,  not  only  when  it  is  hanging  at  rest,  but  also  when- 
ever it  is  raised  or  lowered  with  constant  velocity ;  and  that  the 
tension  is  zero  if  the  stone  is  lowered  with  an  acceleration  equal 
to  that  of  gravity,  as  is  otherwise  evident. 


42.  The  above  formula  will  give  the  tension  T  in  poundals 
(or  dynes),  if  the  mass  m  be  expressed  in  pounds  (or  grammes), 
and  the  accelerations  in  feet  (or  centimetres)  per  second  per 
second. 

In  engineering  practice,  gravitation  measure  is  commonly 
used  for  weights  as  well  as  for  the  forces  that  replace  con- 
straints (tensions,  pressures,  friction,  etc.).  The  engineer  would, 
therefore,  divide  by  g  the  numerical  value  of  T  just  found,  so  as 
to  reduce  it  to  pounds. 

It  should  be  noticed  that  the  general  equations  of  theoretical 
mechanics  are  of  course  independent  of  the  system  of  units 
adopted,  and  that  in  applying  them  to  numerical  examples  it 
is  only  necessary  to  use  one  and  the  same  system  of  ~ units  con- 
sistently throughout.  As  modern  physics  has  settled  upon 
mass  as  a  fundamental  unit  (see  Part  II.,  Art.  68),  regarding  the 
unit  of  force  as  derived  from  and  based  upon  the  unit  of  mass, 
this  absolute  system  will  always  be  adopted  in  this  book,  unless 
the  contrary  be  specified.  In  other  words,  it  will  always  be 
assumed  that  mass  is  expressed  in  pounds  (or  grammes),  and 
consequently  force  in  potmdals  (or  dynes). 

43.  Let  us  next  consider  two  particles,  mlt  mv  connected 
by  a  cord  hung  over  a  vertical  fixed  pulley,  as  in  the  apparatus 
known    as   Atwood's   machine   (Fig.    5).      If   m^  >  m%,  m-^  will 
descend  while  m^  ascends.     The  effective  force  of  the  system 
formed  by  the  two  particles  is  evidently  the  difference  of  the 
weights  of   the  particles,  viz.    W1—  W2  =  (ml  —  m^g>  while  the 
whole  mass  moved  (neglecting  the   mass  of  the   cord  and  of 


22 


KINETICS   OF   A   PARTICLE.  [44. 

Hence,  we  have  for  the  acceleration 


the  pulley)  is 
j  of  the  system, 


This  acceleration  is  constant,  and  the  relations  between  space, 
time,  and  velocity  are  found  just  as  for  a  single  particle  falling 
freely,  except  that  the  acceleration  of  gravity  g 
is  replaced  by  the  fraction  (ml  —  m2)/(m1-\-m2) 
of  g.  It  follows  that  if  the  masses  mlt  m2  be 
selected  nearly  equal,  the  acceleration  will  be 
small,  and  the  motion  can  be  observed  more 
conveniently  than  that  of  a  freely  falling  body. 


W2 


Fig.  5. 


44.  The  tension  T  of  the  cord  is,  of  course, 
the  same  at  every  point  of  the  cord  if,  as  is 
here  assumed,  the  weight  of  the  cord  and  the 
axle-friction  of  the  pulley  be  neglected.  To 
determine  this  tension,  we  have  only  to  consider  either  particle 
separately. 

If    the   cord   be   cut    just    above   mlt   and    the    tension    T 
be   introduced,  the   particle   m1  will  move  like  a  free  particle    , 
under    the    action   of    the   resultant   force     Wl—T=mlg—T./ 
Hence,  as  the  sense  of  the  acceleration/  of  ml  agrees  with  that 


mlg-T 
m1 

Similarly,  we  have  for  the  acceleration  of 
opposite  to  that  of  gy 

m^g-T 


(6) 


Eliminating  /  between  any  two  of  the  equations  (4),  (5),  (6), 
we  find  the  tension 


(7) 


45-] 


RECTILINEAR   MOTION. 


45.  If  the  two  particles  of  Art.  43  move  on  inclined  planes 
intersecting  in  the  horizontal  axis  of  the  pulley  (Fig.  6),  it  is 
only  necessary  to  resolve  the  weights  m^g  and  m^g  into  two 
components,  one  parallel,  the  other  perpendicular,  to  the  inclined 


Fig.  6. 


plane.  If  the  planes  be  smooth,  the  system  formed  by  the 
two  particles  is  made  free  by  introducing  the  normal  reactions 
of  the  planes  which  counterbalance  the  perpendicular  com- 
ponents of  the  weights.  The  effective  force  is  therefore  the 
•difference  of  the  parallel  components,  and  the  acceleration  is 


m-i  sin  01  —m0  sin 

7  — •  L 1 A 

J  — 


(8) 


where  0V  02  are  the  angles  of  inclination  of  the  planes  to  the 
horizon. 

The  tension  T  of  the  connecting  cord  is  again  determined  by 
equating  this  value  of  j  to  the  one  obtained  by  considering 
either  of  the  two  particles  separately.  Thus,  m^  taken  by  itself, 
becomes  free  if  we  introduce  not  only  the  normal  reaction  of 
the  plane,  but  also  the  tension  of  the  string.  This  gives 


(9) 

(10) 


With  0l  =  0z  =  TT/2  the  formulae  (8)  to  (11)  reduce,  of  course, 
to  the  formulae  (4)  to  (7). 


'Similarly,  we  have  for 
i  -—• 

Hence, 


24  KINETICS   OF  A   PARTICLE.  [46. 

46.   Exercises. 

(1)  A  stone  weighing  200  Ibs.  is  raised  vertically  by  means  of  a 
chain  running  over  a  fixed  pulley.     Determine  the  tension  of  the  chain  : 
(a)  when  the  motion  is  uniform ;    (t>)  when  the  motion  is  uniformly 
accelerated  upwards  at  the  rate  of  8  ft.  per  second ;  (c)  when  the  accel- 
eration is  32   ft.  per  second  downwards.     Neglect  the  weight  of  the 
chain  and  the  axle-friction  of  the  pulley. 

(2)  A  railroad  car  weighing  4  tons  is  pushed   by  four  men   over  a 
smooth  horizontal  track.    «If  each  man  exerts  a  constant  pressure  of 
100  pounds,  (a)  what  is  the  velocity  acquired  by  the  car  at  the  end 
of  5  sec.  ?  (^)  what  is  the  distance  passed  over  in  these  5  sec.  ? 

(3)  (a)  Determine  the  constant  force  required  to  give  a  train  of  90 
tons  a  velocity  of  30  miles  an  hour  in  5  min.  after  starting  from  rest. 
(£)  How  far  does  the.  train  go  in  this  time  ?     (c)  If  the  same  velocity 
is  to  be  acquired  at  the  end  of  the  first  mile,  what  must  be  the  tractive 
force  of  the  engine  ? 

(4)  If  in  Atwood's  machine  (Fig.  5)  the  two  masses  are  each  2  lbs.r 
and  an  additional  mass  of  i  oz.  be  placed  on  one  of  these  masses,  how 
long  will  it  take  this  mass  to  descend  6  ft.  ?     (g=  32.2.) 

(5)  If  in  Ex.  (4)  an  additional  mass  of  half  an  ounce  be  placed  on 
each  of  the  2  Ib.  masses,  how  would  the  tension  in  the  cord  differ  from 
the  tension  in  Ex.  (4)  ? 

(6)  Solve  the  problem  of  Art.  45  when  the  inclined  planes  are  rough,. 
the  coefficients  of  friction  being  filt  ^. 

"(?)  A  mass  of  5  Ibs.  rests  on  a  smooth  horizontal  table,  and  has  a 
cord  attached  which  runs  over  a  smooth  pulley  on  the  edge  of  the  table. 
If  a  mass  of  i  Ib.  be  suspended  from  the  cord,  find  the  acceleration  and 
the  tension  of  the  cord. 

(8)  A  sleigh  weighing  500  Ibs.  is  drawn  over  a  horizontal  road,  the 
coefficient  of  friction  being  -fa.     Find  the  pull  exerted  by  the  horses 
when  the  motion  is  uniform. 

(9)  When  the  U.S.S.  Raritan  was   launched   she  was  observed  to 
pass  in  ii  sec.  over  an  incline  of  3°  40',  54  ft.  long.     Find  the  coeffi- 
cient of  friction. 

(10)  A  coaster,  after  coming  down  a  hill,  runs  up  another  hill  a  dis- 
tance of  200  ft.  (from  its  foot)  in  10  sec.,  when  it  stops.     If  the  slope 
of  the  second  hill  be  6°,  find  the  coefficient  of  friction. 


48.]  RECTILINEAR   MOTION.  2£ 

(n)  A  train  of  120  tons  is  running  25  miles  an  hour.  Find  what 
constant  force  is  required  to  bring  it  to  rest  :  (a)  in  3  min.  ;  (b)  in 
half  a  mile. 

(12)  If  it  takes  i  min.  to  coast  down  a  hill  on  a  uniformly  sloping 
road  of  i  mile  length,  and  the  coefficient  of  friction  be  0.02,  what  is  the 
height  of  the  hill  ? 

47.  Kinetic  Energy  and  Work.  The  dynamical  equation  of 
motion,  (i),  Art.  38,  can  often  be  integrated  after  multiplying 
both  members  by  vdt=ds\  this  makes  the  left-hand  member  an 
exact  differential,  viz.  the  differential  of  the  kinetic  energy  \  mv*1, 
while  the  right-hand  member,  Fdst  represents  the  elementary 
work  done  by  the  force  F: 


If  F  be  given  as  a  function  of  s  alone,  this  equation  can  be  inte- 
grated, say  from  the  time  tQ  to  the  time  /.  Denoting  by  s^ 
and  VQ  the  values  of  s  and  v  at  the  time  /0  (Fig.  7),  we  find 


I 

«/* 


Fds.  (12) 


This  equation  gives  the  velocity  v  as  a  function  of  the  distance 
s,  counted  from  the  arbitrary  origin  O.     As  v  =  ds/dt,  a  second 

t0,  V0  t,  V 


fc 

U 


—  -s=s-0— 
Fig.  7. 

integration  will  give  s  as  a  function  of  t.  Examples  of  this 
method  have  been  given  in  Part  I.,  Arts.  109,  117,  119,  121, 
122  ;  it  will  here  only  be  necessary  to  call  attention  to  the 
dynamical  meaning  of  the  quantities  involved. 

48.  The  left-hand  member  of  equation  (12)  represents  evi- 
dently the  increase  in  the  kinetic  energy  of  the  moving  particle, 
while  the  right-hand  member  expresses  the  work  done  by  the 
force  .F.  during  the  passage  of  the  particle  from  the  point  PQ 
to  the  point  P  (see  Part  II.,  Arts.  71,  72).  Hence,  the  meaning 


26  KINETICS    OF   A   PARTICLE.  [49. 

of  the  equation  is  that  the  increase  in  the  kinetic  energy  is  equal 
to  the  work  done  by  the  resultant  force.  This  is  the  principal  of 
work  or  of  kinetic  energy  (or  of  vis  viva)  for  the  case  of  the  rec- 
tilinear motion  of  a  particle. 

Thus,  for  a  falling  body,  F  is  constant  and  equal  to  the  weight 
mg  of  the  body;  hence,  equation  (12)  gives,  if  s  be  counted, 
positive  downwards, 


1  -  \rnvg  =  mg(s — SQ), 

where  the  right-hand  member  represents  the  work  done  by  the 
weight  of  the  body,  i.e.  by  the  attractive  force  of  the  earth 
during  the  fall  of  the  body  through  the  distance  s— SQ. 

For  a  body  thrown  vertically  upwards  with  an  initial  velocity 
v§,  we  have 

J  mi? —±mv<?=  —  mgsy 

if  s  be  counted  from  the  starting  point  and  positive  upwards. 
The  kinetic  energy  here  decreases,  the  initial  kinetic  energy, 
\  mv^,  being,  so  to  speak,  consumed  by  the  work  done  against 
the  force  of  gravity. 

49.   Inclined  Plane.    When  a  particle  of  mass  m  is  moved 
uniformly  up  a  smooth  inclined  plane  from  PQ  to  Pl  (Fig.  8), 

the  work  done  against  gravity 
FV^         is  equal  to  the  work  that  would 
8^^^  have  to  be  done  in  raising  the 

\h  particle  m  through  the  vertical 

height   PPl  of   Pl  above  the 


Po^p        \ —    _j\ * L_      _  jnitia]  point /V     For,  putting 


PQPl=s,  PP^=ht  and   denot- 
ing the  inclination  of  the  plane 
Pig.  s.  to  the  horizon  by  0,  we  have 

•          for  the  work, 

mg-sin  6  •  s=-mg  •  s  sin  Q  =  mg  -  h. 
If  the  plane  be  rough,  the  coefficient  of  friction  being  /*,  the 


Si.]  RECTILINEAR   MOTION.  27 

^**£y£2j»i" 

effective  force  for  motion  upwards  is   =  mg  sin  0  +  yang  cos  6 ; 
hence,  the  work  done  in  moving  the  mass  m  from  PQ  to  Pl  is 


sin  0  4-  prngs  cos  V  =  mg 

where  !=PQP  is  the  horizontal  distance  of  the  final  position  Pl 
from  the  starting  point  PQ.  The  total  work  is,  therefore,  the 
sum  of  the  work  of  overcoming  gravity  through  the  vertical 
distance  h  and  the  work  of  overcoming  friction  through  the 
horizontal  distance  /. 

50.  Work  done  on  a  System  of  Particles.  Let  there  be  given 
any  number  of  particles  of  masses  m-^  m^,  ...  m^  at  the  distances 
.slf  sz,  ...  sn  above  a  fixed  horizontal  plane ;  and  let  these  masses 
be  raised  vertically  against  gravity  so  that  their  distances  from 
the  same  plane  become  s^,  s2',  ...  snf.  The  centroid  of  the 
masses  in  their  original  position  has  a  distance  s  =  2ms/2m 
from  the  fixed  plane,  while  in  the  final  position  it  has  the  dis- 
tance sf=*2msff£m  from  the  same  plane.  It  has,  therefore,  been 
raised  through  a  distance  sr—s.  It  follows  that  the  total  work 
done  in  raising  the  separate  masses,  viz. 


is  equal  to  the  work  that  would  be  done  in  raising  the  total  mass 
through  the  distance  s'  —  s  traversed  by  the  centroid,  i.e.  to 


51.  The  Work  of  a  Variable  Force  is  well  illustrated  by  the 
•expansion  of  gas  or  steam  in  a  cylinder  with  a  movable  piston 
(Fig.  9).  Let  r  be  the  radius  of  the  cylinder,  /  the  pressure  (in 
pounds)  at  any  instant  of  the  gas  per  square  inch  of  surface  ; 
then  the  total  pressure  of  the  gas  on  the  inside  of  the  piston  is 
P  —  rrrr^p  pounds,  and  if  PQ  be  the  pressure  on  the  outside  (say 
the  atmospheric  pressure),  the  effective  force  acting  on  the 
piston  is  F—P—P^  friction  being  neglected. 

The  force  F  is  variable,  since  the  pressure  p  varies  with  the 


28 


KINETICS   OF  A   PARTICLE. 


[52. 


volume   v   occupied  by  the  gas.      This  volume  being   in   the 
present  case  proportional  to  the  distance  s  of  the  piston  from 

the  fixed  base  of  the  cylinder,  the 
force  F  is  a  function  of  s.  The 
variation  of  F  can  therefore  be  rep- 
resented graphically  by  a  curve  hav- 
ing s  for  abscissa  and  F  for  ordinate 
(Fig.  9)  ;  and  the  area  of  this  curve, 
i.e.  the  area  contained  between  the 
curve,  the  axis  of  s,  and  two  ordi- 
nates  whose  abscissas  are  s0  and  s, 

being  given  by  the  integral  j   Fds, 

represents  the  work  done  on  the  piston 
when  ptished  through  the  distance 
s-s0. 


Fig.  9. 


52,  In  the  case  of  a  perfect  gas,  Boyle's  law  gives  the  rela- 
tion pv  =  k,  where  k  is  constant  if  the  temperature  remains  con- 
stant. Hence, 


where  K  and  PQ  are  constants.  This  equation  represents  an 
equilateral  hyperbola,  whose  asymptotes  are  the  axis  of  F  and  a 
line  parallel  to  the  axis  of  s.  For  steam,  the  law  connecting 
pressure  and  volume  is  more  complicated,  but  the  curve  may 
be  taken  as  very  nearly  hyperbolic. 


53.  The  Steam-engine  Indicator  is  an  apparatus  for  measuring  the 
pressure  of  the  steam  in  the  cylinder  and  at  the  same  time  recording  it 
automatically  on  a  drum  revolving  as  the  piston  moves.  Thus,  if  the 
indicator  be  put  in  connection  with  the  interior  of  the  cylinder,  the 
curve  traced  by  the  indicator  has  for  its  abscissas  the  distances  s  of  the 
piston  from  this  end,  and  for  its  ordinates  the  corresponding  pressures 
F  of  the  steam  on  the  inside  of  the  piston. 


56.]  RECTILINEAR   MOTION.  29 

At  the  beginning  of  the  stroke,  steam  is  admitted  and  acts  with  nearly 
constant  pressure  on  the  piston;  the  line 
AB  (Fig.  10)  traced  by  the  indicator  will 
therefore  be  nearly  parallel  to  the  axis  of  s. 
As  soon  as  the  steam  is  shut  off  by  the  slide- 
valve,  the  steam,  being  now  confined  within 
the  cylinder,  begins  to  expand  nearly  accord- 
ing to  the  law  pv  =  const.,  or  Fs  =  const. ; 
the  curve  traced  by  the  indicator  is  therefore 
approximately  an  equilateral  hyperbola  BC,  Fig.  10. 

having  the  axes  as  asymptotes.     When  the 

slide-valve  connects  the  cylinder  with  the  condenser,  a  partial  vacuum  is 
established  behind  the  piston,  and  the  pressure  curve  is  approximately 
a  line  CD,  parallel  to  the  axis  of  F. 

54.  The   area  ABCDO    evidently    represents    approximately    the 
work  of  the  pressure  on  the  inside  of  the  piston  in  one  complete  (for- 
ward and  backward)   stroke.     In  reality,  a  large  number  of  circum- 
stances produce  deviations  from  the  regular  shape  ABCDO,  and  the 
actual  trace,  obtained  by  means  of  an  indicator  for  one  (forward  and 
backward)   stroke,  usually  called  the  indicator  diagram,  forms  a  loop 
somewhat  like  that  indicated  by  the  dotted  curve  in  Fig.  10.     The  area 
of  this  loop,  which  represents  the  work  in  question,  can  readily  be  found 
by  dividing  it  up  into  narrow  rectangular  strips,  or  with  the  aid  of  a 
planimeter. 

55.  The  effective  piston  pressure  is  of  course  the  difference  between 
the  pressures  on  the  two  sides  of  the  piston.     A  diagram  should  there- 
fore be  obtained  for  each  side  of  the  piston ;  from  these  two  diagrams 
the  curve  of  effective  piston  pressure  is  then  derived  by  constructing  the 
curve  whose  ordinates  are  the  differences  of  the  corresponding  pressures 
on  the  two  sides.     By  dividing  the  area  contained  between  this  curve 
and  the  axes  by  the  length  of  the  stroke,  the  average,  or  mean,  piston 
pressure  is  finally  found. 

For  details  the  student  is  referred  to  special  works  on  the  steam 
engine,  such  as  G.  C.  V.  HOLMES,  The  steam  engine,  New  York,  Apple- 
ton,  1887,  pp.  317-345. 

56.  Attractive   and   Repulsive  Forces.     Let   us   consider   the 
motion  of  a  particle  acted  upon  by  a  so-called  central  force,  i.e. 


30  KINETICS   OF   A   PARTICLE.  [57. 

a  force  whose  direction  constantly  passes  through  a  fixed  centre 
(9,  while  its  magnitude  is  a  function  of  the  distance  s  from  the 
centre  alone.  If  the  initial  velocity  be  zero,  or  if  its  direction 
pass  through  the  centre  O,  the  motion  of  the  particle  will  be 
rectilinear,  the  line  of  motion  passing  through  the  centre  of 
force,  O.  The  most  important  special  cases  of  this  kind  have 
been  treated  in  kinematics  (Part  I.,  Arts.  117-124,  176). 

57.  Let  the  force  be  due  to  a  mass  m'  concentrated  at  the 
centre  O,  and  attracting  according  to  Newton's  law  of  the 
inverse  square  of  the  distance  (Part  II.,  Art.  257). 
Counting  the  distances  s  from  the  centre  O  as  origin 
(Fig.  u),  we  have,  for  the  force  acting  on  the  par- 
ticle my 


FN 


where  K  is  a  constant  whose  value  may  be  deter- 
mined, as  indicated  in  Part  I.,  Art.  119,  and  Part  II., 
Arts.  262,  263. 

The  principle  of  kinetic  energy,  equation  (12),  Art. 
lg'     '      47,  gives  at  once 


The  quantity  m' /s,  or  in  absolute  measure  tcm'/s,  is  the 
potential  at  P  due  to  m'  (Part  II.,  Art.  278),  and  /cm'/sQ  is  the 
potential  at  P0  due  to  the  same  mass  m'.  The  increase  in 
kinetic  energy  is,  therefore,  proportional  to  the  decrease  in 
potential. 

The  quantity  /cmm'/s  is  sometimes  called  the  mutual  potential 
of  the  masses  m  and  m' ;  hence,  the  increase  of  the  kinetic 
energy  can  be  said  to  be  equal  to  the  difference  of  the  mutual 
potentials  in  the  final  and  initial  positions. 

The  negative  of   the  mutual  potential  is  designated  as  the 


60.]  RECTILINEAR   MOTION.  3I 

potential  energy  of  the  moving  particle  m.     Denoting  this  by 
V,   and  the  kinetic  energy   by   T,  the  last  equation  becomes 


or  7>  V=  ro  +  ro=const.  ; 

i.e.  the  sum  of  the  kinetic  and  potential  energies  remains  con- 
stant during  the  motion.  This  is  the  principle  of  the  conservation 
of  energy  for  this  particular  problem. 

58.  It  is  easy  to  see  that  the  principle  of  the  conservation 
of  energy  holds  generally  whenever  the  resulting  force  F  is  a 
function  of  the  distance  s  alone. 

Indeed,  if  F=  F(s),  the  principle  of  kinetic  energy  gives 

±mv*-±mv*=  \  F(s)ds;  (13) 

»/«0 

hence,   putting  J  F(s)ds=f(s),   where  f(s)   is   called   the  force- 

function,  or  potential  function,  while  —f(s)  is  the  potential 
energy,  we  have 

\mT?-\mV*=f(s)-f(S^  (14) 

or,  with  the  notation  of  Art.  57, 

=ro+F0  =  const.  (15) 


59.   When  the  resultant  force  F  is  an  attraction  directly  proportional 
to  the  distance  s  from  a  fixed  centre  O,  say 


the  potential  energy  is,  by  Art.  58, 

Hence,  the  principle  of  the  conservation  of  energy  gives 

z/2  -f-  KV  =  const,  j 
or,  if  the  initial  velocity  is  zero  when  s  =  s0, 


60.   Tension  of  an  Elastic  String.     According  to  Hooke's  law,  the 
tension  of  an  elastic  string  is,  within  the  limits  of  elasticity  (i.e.  as  long 


32  KINETICS   OF   A   PARTICLE.  [61. 

as  no  permanent  deformation  is  produced)  ,  directly  proportional  to  the 
extension  or  change  of  length  produced. 

Thus,  let  an  elastic  string,  whose  natural  length  is  /,  assume  the 
length  s  when  its  tension  is  T\  then  Hooke's  law  can  be  expressed  in 
the  form 


where  k  is  a  constant.  To  determine  this  constant  for  a  given  string, 
we  may  observe  the  length  /x  assumed  by  the  string  under  a  known 
tension,  say  the  tension  TI  =  mg,  produced  by  suspending  a  given  mass 
m  from  the  string  (the  weight  of  the  string  itself  being  neglected).  We 
then  have 

71  =£(/i-/). 
Hence,  dividing, 

T  _  s  -  I 
mg     4  -  / 

or,  denoting  by  e  the  extension  l±  —  l  due  to  the  weight  mg, 

T="*f(s-l).  (16) 

61.  By  means  of  this  relation  we  can  determine  the  motion  of  a 
particle  of  mass  m  attached  to  a  fixed  point  O  by  means  of  an  elastic 
string,  if  the  string  be  stretched  and  then  let  go.  We  shall  assume  the 
particle  and  string  to  lie  on  a  smooth  horizontal  table,  so  as  to  eliminate 
the  effect  of  the  weight  of  the  particle. 

The  equation  of  motion  is 


whence,  putting  for  shortness  ^Jg/e  =  K, 

s  =  1+  C\  cos  K/  +  C2  sin  */, 


If  the  initial  length  of  the  string  at  the  time  /=  o  be  s0,  the  constant 
.are  readily  determined,  and  we  find 

s  =  /-f  (.r0  —  /)  cos  K/, 

v  =  —  K  (s0  —  /)  sin*/.  (18) 


62.]  RECTILINEAR   MOTION.  33 

It  should  be  noticed  that  these  equations  hold  only  as  long  as  the 
string  is  actually  stretched,  i.e.  as  long  as  s  >  / ;  the  motion  that  ensues 
when  s  becomes  less  than  /  is,  however,  easily  determined  from  the 
velocity  for  s  =  I. 

62.   Exercises. 

(1)  In  a  steam  engine,  let/=  15  Ibs.  per  square  inch  be  the  mean 
piston  pressure  during  one  stroke,  s  =  4  ft.  the  length  of  the  stroke,  and 
.d=  1.5  ft.  the  diameter  of  the  cylinder,     (a)  What  is  the  work  per 
stroke?     (b)  To  what  height  could  a  mass  of  500  Ibs.  be  raised  by  this 
work? 

(2)  A  train  of  80  tons  starting  from  rest  acquires  a  velocity  of  30 
miles  an  hour  on  a  level  road  at  the  end  of  the  first  mile.     Determine 
the  average  tractive  force  of  the  engine  :   (a)  if  the  frictional  resistances 
te  neglected ;  (b)  if  these  resistances  be  estimated  at  8  Ibs.  per  ton. 
(c)  What  tractive  force  is  required  to  haul  the  same  train  over  a  level 
road  at  a  constant  speed? 

(3)  A  train  of  60  tons  runs  one  mile  with  constant  speed ;  if  the 
resistances  be  8  Ibs.  per  ton,  find  the  work  done  by  the  engine  :   (a)  on 
a  level  track ;   (b)  on  an  average  grade  of  i  % .     (t)  On  a  i  %  grade, 
what  is  the  ratio  of  the  work  done  against  gravity  to  that  done  against 
the  resistances  ? 

(4)  Determine  the  work  expended  in  raising  from  the  ground  the 
materials  for  a  brick  wall  30  ft.  high,  40  ft.  long,  and  2  ft.  thick,  the 
weight  of  a  cubic  foot  of  brickwork  being  112  Ibs. 

(5)  Knowing  that  on  the  surface  of  the  earth  the  attraction  per  unit 
•of  mass  is  g=  32,  find  what  it  would  be  on  the  sun  if  the  density  of  the 
sun  be  \  of  that  of  the  earth,  and  its  diameter  108  times  that  of  the 
earth. 

(6)  Show  that  the  velocity  acquired  by  a  body  in- falling  to  the  sur- 
face of  the  earth  from  an  infinite  distance,  under  the  action  of  the 
earth's   attraction  alone,  would  be  v  =  ^/2gR,  or  about   7   miles  per 
second  (with  R  =  4000  miles). 

( 7)  A  homogeneous  straight   rod,  AB  =  /,   of  constant   density  p, 
attracts  a  particle  P  of  mass  i   according  to  the  law  of  the   inverse 
square  of  the  distance.     The  initial  position  P0  of  P  is  on  AB  produced 
beyond  B,  at  the  distance  BP^  =  SQ,  and  the  initial  velocity  is  zero. 

PART   III — 3 


34  KINETICS   OF   A   PARTICLE.  [62. 

(a)  Determine  the  velocity  v  of  P  at  any  distance  BP  =  s,  and  its 
velocity  v1  at  B.  (b}  How  is  the  solution  to  be  modified  if  the  linear 
mass  BA  extends  from  B  to  infinity  ? 

(8)  A   circular  wire  of  radius  a  and   constant   density  p   attracts, 
according  to  Newton's  law,  a  particle  P  of  mass  i,  situated  on  the  axis 
of  the  circle  ;  i.e.  on  the  perpendicular  to  its  plane  passing  through  the 
centre  O.     If  the  velocity  is  zero  when  the  particle  is  at  the  distance 
OP^  =  s0,  determine  the  velocity  of  the  particle  at  any  distance  s,  and 
show  that  the  motion  is  oscillatory. 

(9)  Determine  the  motion  of  two  free   particles  of  masses  mlf  m2f. 
attracting  each  other  according  to  Newton's  law,  and  starting  at  the 
distance  j0  with  zero  velocity. 

(10)  Show  that  the  motion  of  the  particle  in  Art.  61  is  oscillatory, 
and  that  the  period,  i.e.  the  time  of  one  complete  oscillation,  is 


(n)  A  particle  of  mass  m  is  suspended  from  a  fixed  point  by  means 
of  an  elastic  string  whose  weight  is  neglected.  The  natural  length  of 
the  string  is  /.  Its  length,  when  the  mass  m  is  suspended  at  its  end,  is 
/!•  If  the  particle  be  pulled  down  so  as  to  make  the  length  of  the 
string  =  s0,  and  then  released,  the  particle  will  perform  oscillations. 
Determine  their  period  :  (a)  if  s0  —  /x  <  /x  —  /;  (£)  if  s0  —  ^  >  ^  —  I. 

(12)  The  particle  in  Ex.  (u)  is  raised  through  a  height  h,  so  as  to 
loosen  the  string,  and   then  dropped.     Determine  the  greatest  exten- 
sion of  the  string. 

(13)  An   elastic  string,  whose  natural  length   is  =  /,  is   suspended 
from  a  fixed  point.     A  mass  m±  attached  to  its  lower  end  stretches  it  to 
a  length  /x  ;  another  mass  m2  stretches  it  to  a  length  /2.     If  both  these 
masses  be  attached  and  then  the  mass  m2  be  cut  off,  what  will  be  the 
motion  of  ml  ? 

(14)  A  particle  performs  rectilinear  oscillations  owing  to  a  centre  of 
force  in  the  line  of  motion  attracting  the  particle  with  a  force  directly 
proportional  to  the  distance.     The  motion  of  the  particle  is  impeded 
by  a  resistance  directly  proportional  to  the  velocity.     Investigate  the 
motion. 


65.]  RECTILINEAR  MOTION.  35 

63.  Power.     It  has  been  shown  that  the  time-effect  of  a  force 
is  measured  by  its  impulse  (Art.  2),  while  the  space-effect  is 
measured  by  its  work  (Arts.  47,  48).     In  applied  mechanics  it  is 
of  great  importance  to  take  time  and  space  into  account  simul- 
taneously.     The  time-rate  at  which  work  is  performed  by  a  force 
has  therefore  received  a  special  name,  power.      The  source  from 
which  the  force  for  doing  useful  work  is  derived  is  commonly 
called  the  agent ;  and  it  is  customary  to  speak  of  the  power  of 
an  agent,  this  meaning  the  rate  at  which  the  agent  is  capable 
of  supplying  useful  work. 

64.  The  dimensions  of  power  are  evidently  ML2T~S.     The 
unit  of  power  is  the  power  of  an  agent  that  does  unit  work  in 
unit  time.     Hence,  in  absolute  measure,  it  is  the  power  of  an 
agent  doing  one  erg  per  second  in  the  C.G.S.  system,  and  one 
foot-poundal  per  second  in  the  F.P.S.  system.      As,  however, 
the  idea  of  power  is  of  importance  mainly  in  engineering  prac- 
tice, power  is  usually  measured   in  gravitation  units.     In  this 
case,  the  unit  of  power  is  the  power  of  an  agent  doing  one  foot- 
pound per  second  in  the  F.P.S.  system,  and   one  kilogramme- 
metre  in  the  metric  system. 

A  larger  unit  is  frequently  found  more  convenient.  For  this 
reason,  the  name  horse-power  (H.P.)  is  given  to  the  power  of 
doing  550  foot-pounds  of  work  per  second,  or  550x60  =  33,000 
foot-pounds  per  minute. 

65.  Efficiency  of  Machines.     While  the  principle  of  the  con- 
servation of  energy  was  proved  in  Arts.  57  and  58  only  for  a. 
special  case,  it  is  known  to  be  of  almost  universal  application  to 
the  forces  occurring  in  nature.     Thus,  in  particular  in  the  case 
of  machines  it  is  found  to  be  verified  with  a  degree  of  approxi- 
mation corresponding  to  the  precision  of  the  investigation. 

The  principle  can  here  be  expressed  in  the  form 

W=Wt+Wl9 
if  W  denote   the   total  work   done   by  the  agent  driving  the 


36  KINETICS   OF   A   PARTICLE.  [66. 

machine  (such  as  animal  force,  the  expansive  force  of  steam, 
the  pressure  of  the  wind,  etc.) ;  WQ  the  so-called  lost  or  wasteful 
work  spent  in  overcoming  friction  and  other  passive  resistances 
of  the  machine  ;  and  W^  the  useful  work  done  by  the  machine. 

While  W  and  Wl  allow  of  precise  determination,  it  is  in 
general  difficult  to  determine  WQ  accurately ;  but  it  is  found 
that  the  more  exactly  in  any  given  machine  W§  is  determined, 
the  more  nearly  will  the  equation  W=  WQ  +  Wl  be  fulfilled. 

As  explained  in  Part  II.,  Art.  255,  the  ratio  W^l  W  of 
the  useful  work  to  the  total  work  is  called  the  efficiency  of  the 
machine.  The  term  modulus  is  used  sometimes  for  efficiency. 

66.   Exercises. 

(1)  In  electrical  engineering  a  watt\s  defined  as  the  power  of  doing 
one  joule,  i.e.  io7  ergs,  per  second.     Find  the  relation  between  the  watt 
and  the  horse-power. 

(2)  In  countries  using  the  metric  system  of  weights  and  measures 
the  horse-power  is  defined  as  75  kilogramme-metres  per  second.     Find 
its  relation  to  the  watt  and  to  the  British  horse-power. 

(3)  Find  the  horse-power  of  the  engine  in   Art.  62,  Ex.  (i),  if  it 
make  i  stroke  per  second. 

(4)  The  cylinder  of  a  steam  engine  has  a  diameter  of  15  in. ;  the 
stroke  is  3  ft. ;  the  number  of  strokes  per  minute  is  77  ;  the  mean  press- 
ure of  the  steam  is  40  Ibs.  per  square  inch.     What  is  the  horse-power 
of  the  engine  ? 

(5)  Find  the  horse-power  required  of  the  locomotive  to  haul  a  train 
of  100  tons  at  the  rate  of  30  miles  an  hour,  the  resistances  amounting 
to  8  Ibs.  per  ton :   (a)  on  a  level  road ;  (b)  up  a  i  %  grade ;   (c)  up  a 
2%  grade. 

(6)  How  much  water  can  an  engine  furnishing  50  H.P.  raise  per 
minute  from  the  bottom  of  a  mine  1000  ft.  deep? 

(7)  The  diameter  of  the  cylinder  of  a  steam  engine  is  30  in.;    the 
stroke  4  ft. ;  the  mean  pressure  15  Ibs.  per  square  inch ;  the  number  of 
revolutions  24  per  minute.     If  the  efficiency  of  the  engine  be  f ,   what 
is  the  amount  of  water  raised  per  hour  from  a  depth  of  250  ft.? 


66.]  RECTILINEAR   MOTION.  37 

(8)  In  what  time  would  an  engine  yielding  2  H.P.  perform  the  work 
of  raising  the  brickwork  in  Art.  62,  Ex.  (4)  ? 

(9)  A  shaft  of  8  ft.  diameter  is  to  be  sunk  to  a  depth  of  420  ft. 
through  a  material  whose  specific  gravity  is  2-2.     Determine:   (a)  the 
total  work  of  raising  the  material  to  the  surface  ;  (<£)  the  time  in  which 
it  can  be  done  by  an  engine  yielding  3-5  H.P.  ;  (c)  the  time  in  which 
it  can  be  done  by  4  men  working  in  a  capstan,  if  each  laborer  does 
2500  ft.-lbs.  per  minute,  working  8  hours  per  day. 


38  KINETICS    OF   A   PARTICLE.  [67. 

III.     Free  Curvilinear  Motion. 

I.       GENERAL    PRINCIPLES. 

67.  Let  j  be  the  acceleration  of  a  particle  of  mass  m  at  the 
time  /;  Fthe  resultant  of  all  the  forces  acting  on  the  particle ; 
then  its  equation  of  motion  is  (Art.  35) 

mj=F. 

In  curvilinear  motion  (Fig.  12)  the  direction  of  j  and  F  differs 
from  the  direction  of  the  velocity  v ;  and  the  angle  i/r  between/ 
and  v  varies  in  general  in  the  course 
of  time.  As  shown  in  kinematics  (Part 
L,  Art.  159),  the  acceleration  can  be 
resolved  into  a  tangential  component 
jt=zdv/dt=*d?s/dP  and  a  normal  com- 
ponent jn  =  iP/pt  where  p  is  the  radius 
of  curvature  of  the  path.  Hence,  if  the 
resultant  force  F  which  has  the  direc- 
tion of  j  be  resolved  into  a  tangential 

force  Ft  =  Fcos-*lr,  and  a  normal  force  Fn  =  Fsm^t  the  above 
equation  of  motion  will  be  replaced  by  the  following  two  equa- 
tions : 

m%=F0      mv-  =  Fn.  (i) 

dt  p 

In  the  particular  case  when  the  normal  component  Fn  is  con- 
stantly directed  towards  a  fixed  point  it  is  called  centripetal 
force. 

68.  The  formulae  (i)  show  how  the  force  .F  affects  the  veloc- 
ity of  the  particle  and  the  curvature  of  the  path.      The  change 
of  the  magnitude  of  the  velocity  is  due  to  the  tangential  force 
Ft  alone.      If  this  component  be  zero,  i.e.  if  the  resultant  force 
F  be  constantly  normal  to  the  path,  the  velocity  v  will  remain 
of  constant  magnitude.     The  curvature  of  the  path,   i/p,  and 


7i.]  PRINCIPLE    OF    KINETIC   ENERGY.  39 

hence  the  direction  of  v,  depends  on  the  normal  component  Fn. 
If  this  component  be  zero,  the  curvature  is  zero  ;  i.e.  the  path 
is  rectilinear. 

69.  Instead  of  resolving  the  resultant  force  F  along  the  tan- 
gent and  normal,  it  is  often  more  convenient  to  resolve  it  into 
three  components,  Fcosa  =  X,  Fcos/3  =  Y,  Fcosy  =  Z,  parallel 
to  three  fixed  rectangular  axes  of  co-ordinates  Ox,  Oy,  Oz,  to 
which  the  whole  motion  is  then  referred.  If  x,  y,  z  be  the 
co-ordinates  of  the  particle  m  at  the  time  /,  the  equations  of 
motion  assume  the  form 


Thus,  the  curvilinear  motion  is  replaced  by  three  rectilinear 
motions. 

70.  If  the  components  X,  Y,  Z  were  given  as  functions  of 
the  time  /  alone,  each  of  the  three  equations  (2)  could  be  inte- 
grated separately.     In  general,  however,  these  components  will 
be  functions  of  the  co-ordinates,  and  perhaps  also  of  the  veloc- 
ity and  time.     No  general  rules  can   be  given  for  integrating 
the  equations  in  this  case.     By  combining  the  equations  (2)  in 
such  a  way  as  to  produce  exact  derivatives  in  the  resulting 
equation,  it  is  sometimes  possible  to  effect  an  integration.     Two 
methods  of  this  kind  have  been  indicated  for  the  case  of  two 
dimensions  in  a  particular  example  in  Part  I.,  Art.  232.     We 
now  proceed  to  study  these  principles  of  integration  from  a 
more  general  point  of  view,  and  to  point  out  the  physical  mean- 
ing of  the  expressions  involved. 

71.  The  Principle  of  Kinetic  Energy.      Let  us  combine  the 
equations  of  motion  (2)  by  multiplying  them  by  dx/dt,  dy/dt, 
dz/dt  respectively,  and  then  adding.      The  left-hand  member  of 
the  resulting  equation  will  be  the  derivative  with  respect  to  /  of 


40  KINETICS   OF   A   PARTICLE.  [72 

We  find,  therefore, 

dx 

dt 

or,  multiplying  by  dt  and  integrating, 

dx+  Ydy  +  Zds),  (3) 

where  VQ  is  the  initial  velocity. 

The  left-hand  member  represents  the  increase  in  the  kinetic 
energy  of  the  particle  ;  the  right-hand  member  represents  the 
work  done  by  the  resultant  force  ;  and  equation  (3)  expresses 
the  equality  between  the  work  done  and  the  change  in  the 
kinetic  energy,  that  is,  the  principle  of  work  or  of  kinetic 
energy  for  the  curvilinear  motion  of  a  particle  (comp.  Art. 
47).  Sometimes  the  name  principle  of  vis  viva  is  given  to 
this  proposition,  the  term  vis  viva,  or  living  force,  meaning  the 
same,  as  kinetic  energy,  or,  in  older  works,  twice  the  kinetic 
energy. 

72.  The  principle  of  work  can  be  deduced  still  more  directly 
from  the  equations  (i).  Multiplying  the  former  of  these  equa- 
tions by  vdt=ds,  we  find 

d(  J  mi?)  =  Fds  cos  1^  ; 
hence,  integrating, 

=       Fds  cos  ^,  (4) 


where  z/0  is  the  velocity  of  the  particle  at  the  place  specified  by 
s0  (comp.  Part  II.,  Art.  72). 

73.  The  principle  of  kinetic  energy  gives  a  first  integral  of 
the  equations  of  motion  whenever  the  integration  indicated  in 
the  right-hand  member  of  (3)  or  (4)  can  be  performed.  We 
proceed  to  investigate  under  what  conditions  this  integration 
becomes  possible. 

In  the  most  general  case  the  components  X,  Y,  Z,  in  (3),  as 
well  as  the  tangential  force  Fcos  t/r  in  (4),  are  functions  of  the 


75-]  PRINCIPLE   OF   KINETIC   ENERGY.  4! 

co-ordinates  x,  y,  z,  of  the  velocity,  i.e.  of  the  time-derivatives 
of  x,  y,  z,  and  of  the  time  t.  If  the  motion  of  the  particle  were 
completely  known,  that  is,  if  we  knew  its  position  at  every 
instant,  the  co-ordinates  would  be  known  functions  of  the  time, 
say 


By  differentiation  the  velocities  vx  =  dxjdt,  vy  =  dy/dt,  vg=dz/dt 
could  be  found  ;  and,  substituting  in  (3),  the  integral  would 

assume  the  form  I     <j>(t}dty  so  that  the  work  could  be  determined 

*AO 

by  evaluating  this  integral.  As,  however,  the  motion  of  the 
particle  is  generally  not  known  beforehand,  this  motion  being 
just  the  thing  to  be  determined,  the  integral  cannot  be  evaluated 
in  the  most  general  case. 

74.    If  the  forces  acting  on  the  particle  depend  only  on  the  posi- 
tion of  the  particle,  i.e.  if  X,  Y,  Z  are  functions  of  x,  y,  z  alone, 

the  integral   I     (Xdx  +  Ydy  -f  Zdz)  can  be  determined  whenever 

the  path  of  the  particle  is  given.  For  the  equations  of  the  path, 
say 

*  *)  =°>  fJi*>  y>  *)  =°> 


make  it  possible  to  eliminate  two  of  the  three  variables  x,  yy  z 
from  under  the  integral  sign,  or  to  express  all  three  in  terms  of 
a  fourth  variable.  In  either  case  the  function  under  the  integral 
sign  becomes  a  function  of  a  single  variable,  and  the  work  of 
the  forces  can  be  found. 

75.  If  the  forces  are  such  as  to  make  the  expression 
Xdx  -f  Ydy  +  Zdz  an  exact  differential,  say  dU,  the  integration 
can  evidently  be  performed  without  any  knowledge  of  the  path  of 
the  particle  between  its  initial  and  final  positions.  In  this  case 
equation  (3)  becomes 


42  KINETICS   OF   A   PARTICLE.  [76. 

UQ  being  the  value  of  £7  at  the  initial  position,  where  v  =  v§.  As 
most  of  the  forces  occurring  in  nature  are  of  this  character,  this 
particular  case  is  of  great  importance,  and  deserves  careful 
study. 

76.  The  expression  Xdx  +  Ydy  +  Zdz  will  be  an  exact  differ- 
ential whenever  there  exists  a  function  U  of  the  co-ordinates 
x,  y,  z  alone  (i.e.  not  involving  the  time  or  the  velocities  ex- 
plicitly), such  that 


Z.  (6) 

dx  By  dz 

If  these  conditions  are  fulfilled,  we  have  evidently 

Xdx+  Ydy  +  Zdz  =  dU. 

The  function  £7  is  called  the  force-function,  and  forces  for  which 
a  force-function  exists  are  called  conservative  forces. 

Hence,  if  the  forces  acting  on  a  particle  are  conservative,  in 
other  words,  if  they  have  a  force-function,  the  principle  of  work 
gives  a  first  integral  of  the  equations  of  motion. 

77.  The  conditions  (6)  for  the  existence  of  a  force-function  U 
can  be  put  into  a  different  analytical  form  which  is  frequently 
useful.  Differentiating  the  second  of  the  equations  (6)  with 
respect  to  z,  the  third  with  respect  to  y,  we  find 


dydz       dz      dzdy     dy 

whence  dY/dz  =  dZ/dy.  If  we  proceed  in  a  similar  way  with 
the  other  equations  (6),  it  appears  that  they  can  be  replaced  by 
the  following  conditions  : 


dz       dy      dx       dz       dy        dx 

It  is  shown  in  works  on  the  differential  calculus  and  dif- 
ferential equations  that  these  equations  (7),  or  the  equations  (6), 
which  are  equivalent  to  them,  are  not  only  the  sufficient,  but 
also  the  necessary,  conditions  that  must  be  fulfilled  to  make 
Xdx+  Ydy  +  Zdz  an  exact  differential. 


:8o.]  PRINCIPLE    OF   KINETIC   ENERGY.  43 

78.  The  dynamical  meaning  of  the  existence  of  a  force-function 
U  lies  mainly  in  the  fact  that,  if  a  force-function  exists,  the  work 
done  by  the  forces  as  the  particle  passes  from  its  initial  to  its 
final  position  depends  only  on  these  positions,  and  not  on  the 
intervening  path.     This  is  at  once  apparent  from  equation  (5), 
in  which  U—  (70  represents  this  work. 

It  follows  that  the  work  of  conservative  forces  is  zero  if  the 
particle  returns  finally  to  its  original  position,  that  is,  if  it 
describes  a  closed  path,  provided  that  the  force-function  U  is 
.single-valued,  an  assumption  which  will  here  always  be  made. 

79.  In   the  case  of  central  forces   inversely  proportional  to 
the  square  of  the  distance,  for  which  a  force-function  can  always 
be  shown  to  exist    (see    Part    II.,   Arts.    278-281),    the   force- 
function  is  usually  called  the  potential.     The  negative  of  the 
force-function,  say 

v=-u, 

is  called  the  potential  energy.  If  this  quantity  be  introduced, 
and  the  kinetic  energy  be  denoted  by  T  (as  in  Art.  57),  the 
equation  (5)  assumes  the  form 

T+  v=  r0+  ra>  (8) 

which  expresses  the  principle  of  the  conservation  of  energy  for  a 

particle  :  the  total  energy,  i.e.  the  sum  of  the  kinetic  and  potential 
energies,  remains  constant  throughout  the  motion  whenever  there 
exists  a  force-function.  In  other  words,  whatever  is  gained  in 
kinetic  is  lost  in  potential  energy,  and  vice  versa. 

80.  The  name  force-function  is  due  to  Sir  William  Rowan  Hamilton. 
Some  authors  use  it  for  V—  —  U,  and  not  for  U.     With  regard  to  the 
term  potential,  the  usage  is  still  less  settled.     Some  writers  use  it  for  U, 
others  for  —  U,  nor  is  its  use  always  restricted  to  Newtonian  forces. 
Green  was  the  first  to  give  the  name  potential  function  to  the  function 
U\  Gauss  brought  the  expression  potential  into  common  use.     Clausius 
uses  "  potential  function  "  for  what  is  called  above  "  potential,"  reserv- 
ing the  latter  name  for  the  potential  of  a  system  on  another  system,  or 
•on  itself.      He  also  uses  the  term  ergal  for  what  is  called  above  "  poten- 
tial energy."     Several  writers  have  followed  him  in  this  terminology. 


44 


KINETICS   OF   A   PARTICLE. 


[81. 


81.  As  the  force-function  U  is  a  function  of  the  co-ordinates 
x,  y,  2  alone,  an  equation  of  the  form 

V=c,  (9) 

where  c  is  a  constant,  represents  a  surface  which  is  the  locus 
of  all  points  of  space  at  which  the  force-function  has  the  same 
value  c.  By  giving  to  c  different  values,  a  system  of  surfaces  is 
obtained,  and  these  surfaces  are  called  level,  or  equipotential, 
surfaces. 

82.  The  values  of  the  derivatives  of  U  at  any  point  P(x,  y,  2) 
are  proportional  to  the  direction-cosines  of  the  normal  to  the 
equipotential  surface  (9)  at  P.     But,  by  (6),  they  are  also  pro- 
portional to  the  direction-cosines  of  the  resultant  force  -Fat  this 
point.     It  follows  that  the  resultant  force  F  at  any  point  P  is 
always  normal  to  the  equipotential  surface  passing  through  P. 

If  the  equation  of  the  equipotential  surfaces  be  given,  the 
resultant  force  F  at  any  point  (xt  yy  2)  is  readily  found,  both  in 
magnitude  and  direction,  from  its  components  (6) : 

fdU\*  ,  /d£A2 

-H-T—   +{•£-]  •          (I0) 

\  ox  j       \  oy  J       \  02  J 

83.  As  the  particle  moves  in  its  path  from  any  point  P  to  an 
infinitely  near  point  P1  (Fig.  13),  it  passes  from  one  equipoten- 
tial surf  ace  (7=cto  another  U=c'. 
Its  velocity  meets  these  surfaces 
at  a  varying  angle,  while  its  ac- 
celeration, which  has  the  direc- 
tion of  the  resultant  force  F,  is 
always  normal  to  these  surfaces. 
The  work  done  by  F  as  the  par- 
ticle moves  from  P  to  P'  is 

Fds  cos  (F,  ds)  =  Fdnt 
i,  P"  being  the  intersection  of  the 


where  PP' =  ds  and  />/>"=, 


normal  at  P  with  the  equipotential  surface  passing  through  P1. 
Hence,  by  Art.  72, 

(11) 


85.]  PRINCIPLE   OF   KINETIC   ENERGY.  45 

The  normal  distance  PP"  =  dn  between  two  equipotential 
surfaces  is  therefore  inversely  proportional  to  the  force  F. 

It  also  appears  that  whenever  the  particle  in  its  path  returns 
to  the  same  equipotential.  surface,  the  work  done  by  F  is  zero, 
.and  hence,  by  (5),  the  velocity  assumes  the  initial  value  VQ. 

84.  Let  a,  /3,  7  be  the  direction  cosines  of  F  at  any  point 
P  ;  X,  ft,  v  those  of  any  straight  line  s  drawn  through  P  ;  and 
let  $  be  the  angle  between  /^and  s,  so  that  cos  (f>  =  a 
Then  the  projection  of  Fon  s  is 


or,  since  by  (6)  aF=d(7/dx,  $F=dU/dy,  yF= 


'~  dx  ds      dy  ds      dz  ds~  ds 

i.e.  the  projection  of  the  resultant  force  on  any  direction  is  the 
derivative  of  the  force-function  with  respect  to  that  direction. 

This  follows  also  from  the  equations  (6),  since  the  directions 
of  the  axes  are  arbitrary. 

If  s  be  taken  tangent  to  the  equipotential  surface  passing 
through  P,  we  have  Fs=dU/ds  =  o  ;  if  it  be  taken  normal  to  this 
surface,  we  find  F8  =  F—dU/dn,  which  agrees  with  (11). 

85.   The  force-function  U  determines,  as  has  been  shown,  a 
system  of  equipotential  surfaces    U=  const.      Starting  from  a 
point  P  on  one  of  these  surfaces,  say  U=c  (Fig.  14),  let  us  draw 
through  P  the  direction  of  the  re- 
sultant force,  which  is   normal  to 
the  surface   U=c  (Art.   82).      Let 
this  direction  intersect  in  P1  the 
next  surface,   L7=c'.      At  P1  draw 
the   normal   to    U=c',   and   let   it 
intersect  the  next  surface,  U=c", 
in  P".     Proceeding  in  this  way,  we 
obtain  a  series  of  points  P,  P1,  P", 
JP'",  ...,  which  in  the  limit  will  form  a  continuous  curve  whose 


46  KINETICS    OF   A   PARTICLE.  [36.. 

direction  at  any  point  coincides  with  the  direction  of  the  result- 
ant force  at  that  point.      Such  a  line  is  called  a  line  of  force. 

The  lines  of  force  evidently  form  the  orthogonal  system  ta 
the  system  of  equipotential  surfaces.  The  differential  equations 
of  the  lines  of  force  are  therefore  : 

dx  _  dy  _  dz 

WJ~WJ~W  (13) 

dx       dy       82 

86.    Exercises. 

(1)  Show  that  a  force-function   exists  when  the  resultant  force  is 
constant  in  magnitude  and  direction. 

(2)  Find  the  force-function  in  the  case  of  a  free  particle  moving 
under  the  action  of  the  constant  force  of  gravity  alone  (projectile  in 
vacua}  ;  determine  the  equipotential  surfaces  and  the  potential  energy. 

(3)  Show  the  existence  of  a  force-function  when  the  direction  of  the 
resultant  force  is  constantly  perpendicular  to   a  fixed  plane,  say   the 
.xy-plane,  and  its  magnitude  is  a  given  function /(z)  of  the  distance  z 
from  the  plane. 

(4)  Find   the   force-function,   the   equipotential   surfaces,    and   the 
kinetic  energy  when  the  force  is  a  function  f(f)  of  the  perpendicular 
distance  r  from  a  fixed  line,  and  is  directed  towards  this  line  at  right 
angles  to  it. 

(5)  Show  that  a  force-function  always  exists  for  a  central  force,  i.e. 
a  force  passing  through  a  fixed  point  and  depending  only  on  the  dis- 
tance from  this  point. 

(6)  Show  the  existence  of  a  force-function  when  a  particle  moves 
under  the  action  of  any  number  of  central  forces. 

(7)  A  homogeneous  sphere  of  mass  m(  attracts  a  free  particle  P  of 
mass  m  with  a  force  F=  urnm1  /r*,  where  K  is  a  constant,  and  r—  OP  is 
the  distance  of /'from  the  centre  of  the  sphere.     Show  that  the  poten- 
tial is  V—  —  Kmm1 /r,  and  that  the  equipotential  surfaces  are  spheres 
whose  common  centre  is  at  O. 

(8)  In  Ex.  (7),  assume  Kmm'=  i,  and  draw  the  intersections  of  the 
equipotential  surfaces  with  a  plane  passing  through  O,  from  r  =  i  centi- 
metre to  r  =  2  centimetres,  with  a  difference  of  potential  =  T^. 


8;.]  PRINCIPLE   OF   AREAS.  47 

(9)  Two  spheres,  whose  masses  are  as  i  to  2,  attract  a  particle  of 
mass  i  according  to  Newton's  law  ;  the  distance  of  the  centres  of  the 
spheres  is  =  4.     Construct  the  equipotential   lines  in  a  plane  passing 
through  the  centres,  by  first  constructing  the  equipotential  lines  for  each 
sphere  separately,  and  then  joining  the   points  of  intersection'  whose 
potential  is  the  same. 

(10)  A  particle  of  mass  m  is  subject  to  the  force  of  gravity  and 
to  the  actions  of  two  fixed  centres   C\,  C2,  one  attracting  with  a  force 
inversely  proportional  to  the  square  of  the  distance,  the  other  repelling 
with  a  force  directly  proportional  to  the  distance.      Find  the  equipoten- 
tial surfaces. 

87.  The  Principle  of  Angular  Momentum  or  of  Areas.      Let  us 

begin  with  the  case  of  plane  motion,  the  equations  of  motion 
being 

d'2 


If  we  combine  these  equations  by  multiplying  the  former  by  y, 
the  latter  by  x,  and  subtracting  the  former  from  the  latter,  we 
find 

-  mxd^-myd^=xY-yX.         •  (14) 

The  right-hand  member  is  the  moment   (with  respect    to  the 
origin)  of  the  resultant  force  F  whose  components  are  X,  F(see 
Part  II.,  Art.  91),  while  the  left-hand  member  is  an  exact  deriva- 
tive, viz.  the  derivative  with  respect  to  the  time  of 
mxdy/dt  —  mydx/dt, 

as  is  easily  verified  by  differentiating  this  quantity.  The  result 
can  therefore  be  written  in  the  form 


and  gives,  if  multiplied  by  dt  and  integrated, 

(16) 


dt  dt 

These  equations  express  the  principle  of  angular  momentum,  or 
of  areas,  for  plane  motion. 


48 


KINETICS   OF   A   PARTICLE. 


[88. 


88.  The  name  principle  of  areas  is  due  to  the  kinematical 
meaning  of  the  left-hand  member  (comp.  Part  I.,  Arts.  227-232). 
As  xdy—ydx  represents  twice  the  infinitesimal  sector  described 
by  the  radius  vector  of  the  point  (x,  y)  during  the  element  of 
time,  the  quantity  xdy/dt—ydx/dt  is  twice  the  sectorial  velocity 
about  the  origin.      Introducing  polar  co-ordinates  by  putting 
jr=r  cos  6,  y=rs'm  0,  we  have  xdy—ydx^r^dQ,  and  denoting  by 
61  the  sector  described  in  the  time  /, 

dt       dt    y  dt~      dt 

The  kinematical  meaning  of  equation  (15),  after  dividing  it 
by  m,  can  therefore  be  stated  as  follows  :  the  time-rate  of  change 
of  twice  the  sectorial  velocity  about  any  point  is  equal  to  the 
moment  of  the  acceleration  about  the  same  point. 

89.  The   dynamical   meaning   of  equation    (15)  appears   by 
considering  that  mdx]dt,  mdy/dt  are  the  components  of  the 

momentum  mv  of  the  moving 
particle  (Fig.  15).  The  prod- 
uct mvp  of  the  momentum  and 
its  perpendicular  distance  from 
the  origin  is  called  the  moment 
of  momentum,  or  the  angular 
momentum,  of  the  particle  about 
the  origin. 

It  appears  from  Fig.  1 5  that 
we  have 

rv          dx 


Fig.  15. 


\ 


The  angular  momentum  is  evidently  nothing  but  twice  the 
sectorial  velocity  multiplied  by  the  mass,  just  as  momentum  is 
linear  velocity  times  mass. 

The  dynamical  meaning  of  equation  (15)  can  therefore  be 
expressed  as  follows  :  the  time-rate  of  change  of  angular  momen- 


90.]  PRINCIPLE    OF   AREAS.  49 

turn  about  any  fixed  point  is  equal  to  the  moment  of  the  resultant 
force  about  the  same  point. 

90.   The  most  important  case  in  which  the  integration  in  (16) 
can  be  performed  is  the  case  when 

xY-yX=o,  (17) 

which  evidently  means  that  the  direction  of  the  resultant  force 
F  passes  through  the  origin.  If  this  condition  be  fulfilled, 
equation  (16)  reduces  to  the  form 

dy          dx 

mxdt~myTt=c' 

where  c  is  a  constant  of  integration  to  be  determined  from  the 
initial  position  and  velocity. 

Kinematically,  this  equation  means  that  the  sectorial  velocity 
remains  constant.     It  can  be  put  into  the  form 

dS 


dt     2m 
whence,  by  integration,  we  find 

S-S^(t-t^.  (19) 

Hence,  if  the  acceleration  passes  constantly  through  a  fixed 
point,  the  sector  S  —  S0  described  about  this  point  in  any  time  t  —  10 
is  proportional  to  this  time. 

This  is  the  principle  of  the  conservation  of  area  for  plane 
motion. 

Dynamically,  equation  (18)  means  that  if  the  resultant  force 
passes  constantly  through  a  fixed  point,  the  angular  momentum 
about  this  point  remains  constant.  The  proposition  can  also  be 
called  the  principle  of  the  conservation  of  angular  momentum. 

If  VQ  be  the  initial  velocity,  /0  the  perpendicular  to  VQ  from 
the  fixed  point,  equation  (18)  can  be  written  in  the  form 


PART   III  —  4 


5O  KINETICS   OF   A   PARTICLE.  [91. 

91.    In  the  general  case  of  three  dimensions  any  two  of  the 
equations  of  motion, 


can  be  combined  by  the  method  of  Art.  87,  and  we  find  thus 


—   -- 


d^x  d^z     d  (      dx  dz\        ,.       ^  . 

'Tf-^a-^ir**-*)-1*-**      (2I) 

^fd^dx\=^ 
dt\       dt  dt) 

The  expression  xdy—ydx  now  represents  the  projection  on 
the  ^/-plane  of  the  infinitesimal  sector  described  by  the  radius 
vector  of  the  particle  during  the  time  dt\  similarly,  ydz—zdy 
and  zdx—xdz  are  the  projections  of  the  same  sector  on  the 
planes  yz  and  zx,  respectively. 

92.  The  right-hand  members  of  the  equations  (21)  are  easily 
seen  to  represent  the  moments  of  the  resultant  force  about  the 
axes  of  x,  y,  z,  if  it  be  remembered  that  the  moment  of  a  force 
with  respect  to  an  axis  is  the  moment  of  its  projection  on  a 
plane  perpendicular  to  the  axis  about  the  point  of  intersection 
of  the  axis  with  the  plane  (Part  II.,  Art.  213).  If  the  moment 
of  the  momentum  mv  of  the  particle  be  defined  in  the  same 
way,  the  quantities  mydzjdt  —  mzdy/dt,  mzdx/dt  —  mxdz/dt, 
mxdy/dt—mydx/dt  are  the  moments  of  momentum,  or,  as  they 
are  also  called,  the  angular  momenta,  about  the  three  axes  of 
co-ordinates  Ox,  Oy,  Oz. 

As  the  axes  are  arbitrary,  the  equations  (21)  express  the 
statement  that  the  moment  of  the  resultant  about  any  fixed  axis 
is  equal  to  the  time-rate  of  change  of  the  angular  momentum 
about  the  same  axis. 


94-]  PRINCIPLE   OF  AREAS.  5! 

93,  The  most  important  case  of  the  application  of  the  equa- 
tions (21)  arises  when  one  or  more  of  the  conditions 

yZ-zY=o,    zX-xZ=v,    xY-yX=o  ,(22) 

are  fulfilled.  The  first  of  these  conditions  means  that  the  pro- 
jection of  the  resultant  force  on  the  jAsr-plane  passes  always 
through  a  fixed  point,  viz.  the  origin  of  co-ordinates  ;  or  what 
amounts  to  the  same  thing,  that  the  resultant  force  always 
intersects  the  axis  of  x.  Similarly,  the  second  condition  means 
that  the  resultant  intersects  the  axis  of  y.  Hence,  if  both  these 
conditions  are  fulfilled,  the  resultant  force  passes  constantly 
through  a  fixed  point,  the  origin  of  co-ordinates. 

It  follows  that  if  any  two  of  the  conditions  (22)  are  fulfilled, 
the  third  must  also  be  fulfilled.  This  is  also  evident  ana- 
lytically, as  any  one  of  the  three  equations  can  be  derived  from 
the  two  others. 

94,  If  the  conditions  (22)  are  fulfilled,  the  integration  of  the 
equations  (21)  gives 

dz  dy 

= 


dy  dx 

drmyTt=c» 

where  cv  c^  CQ  are  constants  depending  on  the  initial  condi- 
tions. These  equations  express  the  proposition  that  if  the 
resultant  force  passes  constantly  through  a  fixed  point,  the  angular 
momentum  about  any  axis  passing  through  this  point  remains 
constant. 

Multiplying  the  equations  (23)  respectively  by  x,  y,  z  and 
adding,  we  find 

°>  (24) 


which  is  the  equation  of  a  plane  passing  through  the  origin. 
As  the  co-ordinates  x,  y,  z  of  the  moving  particle  fulfil  this 


52  KINETICS   OF   A   PARTICLE.  [95. 

equation  independently  of  the  time,  it  follows  that  the  motion  is 
necessarily  plane  whenever  the  conditions  (22)  are  satisfied.  The 
constants  of  integration  cv  c2,  c3  are  evidently  proportional  to 
the  direction-cosines  of  the  normal  to  the  plane  of  motion. 

95.    If  the  equations  (23)  be  written  in  the  form 

\ydz- zdy  _     f     \  zdx—xdz  _    ,     \xdy-ydx_     , 
2        dt  1 '    2        dt  2 '    2        dt 

they  show  that  the  projections  of  the  motion  on  the  three 
co-ordinate  planes  have  constant  sectorial  velocities  <:/,  c%,  cs' ; 
hence,  the  sectorial  velocity  of  the  motion  itself  is  constant,  viz. 

dS       / 


It  follows  in  this  case  that  the  sector  S—SQ,  described  during 
the  time  t—t^  is  proportional  to  this  time : 


5-50=^'2+f2'2+V2('-'o)-  (25) 

96.  'Exercises. 

(1)  A  particle  of  mass  m  is  attracted,  according  to  Newton's  law,  by 
a  mass  m'  concentrated  at  a  fixed  point  O.     If  x0,  y0,  ZQ  be  the  initial 
co-ordinates,  and  x0,  y0,  ZQ  the  initial  velocities  of  the  particle,  find  the 
equation  of  the  plane  in  which  it  moves,  and  show  that  this  plane  passes 
through  O  and  the  initial  velocity. 

(2)  A  particle  is  attracted  by  n  fixed  centres,  whose  forces  are  directly 
proportional  to  the  masses  of  the  centres  and  to  the  distances  from 
them.     Show  that  there  is  one  position  of  equilibrium  for  the  particle, 
and  that  the  motion  takes  place  as  if  the  total  mass  of  all  the  centres 
were  concentrated  at  this  point.     Find  also  the  equation  of  the  plane  of 
the  motion. 

(3)  A  particle  is  acted  upon  by  a  central  force,  i.e.  by  a  force  whose 
direction  passes  through  a  fixed  point,  and  whose  magnitude  is  a  func- 
tion of  the  distance  from  this  point,  say  F=mf(r).     Show  that  the 
path  is  a  plane  curve,  and  find  the  equation  of  the  plane  of  the  motion. 

(4)  The  equation  (15)  can,  by  Art.  89, be  written  d(mvp) /dt=xY—yX. 
Show  that  the  two  terms  of  d(mvp}/dt—  mpdv/dt  +  mvdp/dt  are  equal 


f 


99-]  D'ALEMBERTS    PRINCIPLE.  53 

respectively  to  the  moments  of  the  tangential  and  normal  components 
of  the  resultant  force  F. 

97.  The  Principle  of  d'Alembert.     Let  us  consider  a  particle 
of  mass  m  moving  under  the  action  of  any  forces  Fv  F%,  •••  Fn, 
whose  resultant  is  F.     The  total  acceleration  j  of  the  particle 
has  the  components    d*x/dfi,    d^y/dP,    d^z/dP   parallel  to  the 
rectangular  axes  Ox,  Oy,  Oz.      If  the  forces  Fv  F2,  •••  Fn  be 
imagined  removed,  a  force  equal  to  mj  would  be  required  to  give 
the  particle  the  same  acceleration  j  that  it  had  under  the  action 
of  the  forces  Fv  F2,  •••  Fn.    This  fictitious  force,  mj,  whose  com- 
ponents are  md^x/dt^,  md^y/dfi,  md^z/dfi,  is  called  the  effective 
force.    For  the  sake  of  distinction,  the  forces  Fv  F2,  •>•  Fn,  which 
actually   produce   the   motion,  are   called   the  impressed  forces 
(comp.  Art.  36). 

98.  The  ordinary  equations  of  motion  of  a  particle, 

ir       d^z      ~  /  ^ 

Y'  mW*=Z'  (26) 

where  X,  Y,  Z  are  the  components  of  the  resultant  F  of  the 
impressed  forces,  express  merely  the  equality  between  the 
effective  force  mj  and  the  resultant  impressed  force  F.  It  fol- 
lows that,  if  the  reversed  effective  force  —  mj,  or  its  components, 
—m&xfdP)  —  md^y/dfi,  —  md^z/dfi,  be  combined  with  the 
impressed  forces  Fp  F2,  •••  Fn,  we  have  a  system  in  equilibrium. 
This  is  the  fundamental  idea  of  d'Alembert's  principle. 

99.  The  reversed  effective  force,  —  mj,  is  sometimes  called  the 
force  of  inertia  of  the  particle.  To  understand  the  idea  underlying 
this  expression,  imagine  the  impressed  forces  to  be  removed,  and  then 
push  the  particle  with  the  hand  so  as  to  give  it  the  same  motion  that  it 
had  under  the  action  of  the  impressed  forces.  The  pressure  of  the 
hand  on  the  particle  must  at  every  instant  be  equal  to  the  resultant  F, 
or  to  the  effective  force  mj,  while  the  equal  and  opposite  pressure  of 
the  particle  on  the  hand  represents  the  force  of  inertia.  It  must,  how- 
ever, be  clearly  understood  that  this  force  of  inertia,  or  inertia- resist- 
ance, is  a  force  exerted  on  the  hand  and  not  on  the  particle. 


54  KINETICS   OF    A   PARTICLE.  [100. 

100.  Owing  to  the  fact  that,  by  combining  with  the  impressed 
forces  the  reversed  effective  force,  we  obtain  at  any  given  in- 
stant a  system  in  equilibrium,  it  becomes  possible  to  apply  to 
kinetical  problems  the  statical  conditions  of  equilibrium. 

Since  in  the  case  of  a  single  particle  the  forces  are  all  con- 
current, the  conditions  of  equilibrium  are  obtained  by  equating 
to  zero  the  sums  of  the  components  of  the  forces  along  each 
axis.  This  gives 


and  these  are  the  ordinary  dynamical  equations  of  motion  (see 
(26),  Art.  98). 

101.  The  conditions  of  equilibrium  of  a  system  of  forces  can 
also  be  expressed  by  means  of   the  principle  of  virtual  work 
(Part  II.,  Art.  239).     Thus,  let  &r,  Sj,  8z  be  the  components  of 
any  virtual  displacement  Bs  of  the  particle  ;  then  the  principle 
of  virtual  work  applied  to  our  system  of  forces  gives  the  single 
condition 

0'  (27) 

which  is  of  course  equivalent  to  the  three  equations  (26)  on 
account  of  the  abitrariness  of  the  displacement  Bs. 

The  equation  (27),  which  may  also  be  written  in  the  form 

(28) 

expresses  d'Alembert's  principle  for  a  single  particle  :  for  any 
virtual  displacement  the  sum  of  the  virtual  works  of  the  im- 
pressed forces  is  equal  to  that  of  the  effective  force. 

102.  The  advantage  of  using  the  equations  of  motion  in  the 
form  given  to  them  by  d'Alembert  arises  mainly  from  the  appli- 
cation of   the   principle  of   virtual  work  which   thus  becomes 


I03-]  D'ALEMBERTS    PRINCIPLE.  55 

possible ;  this  will  be  seen  more  clearly  later  on,  in  the  treat- 
ment of  constrained  motion.  For  the  present  it  may  suffice 
to  notice  that,  if  the  actual  displacement  ds  of  the  particle  in 
its  path  be  selected  as  the  virtual  displacement  &s,  equation  (28) 
becomes 


This  is  the  equation  of  kinetic  energy  (Art.  71);  for  the  left- 
hand  member  is  the  exact  differential  d(\mv*)  of  the  kinetic 
energy,  while  the  right-hand  member  represents  the  element 
•of  work  of  the  impressed  forces. 

In  the  particular,  but  very  common,  case  of  conservative  im- 
pressed forces,  the  right-hand  member  is  likewise  an  exact 
differential  dU\  hence,  in  this  case  a  first  integration  can  at 
-once  be  performed,  and  we  find,  as  in  Art.  75, 

U-  UQ.         (30) 

103.  There  is  an  essential  distinction  between  the  principle  of 
-d'Alembert  on  the  one  hand,  and  the  principles  of  kinetic  energy  and 
of  areas  on  the  other.  D'Alembert's  principle  merely  gives  a  con- 
venient form  and  interpretation  to  the  dynamical  equations  of  motion, 
through  the  application  of  the  principle  of  virtual  work ;  but  it  does 
not  show  how  to  integrate  these  equations. 

The  principle  of  kinetic  energy  and  the  principle  of  areas  are  really 
-methods  for  integrating  the  equations  of  motion  under  certain  con- 
ditions. If  we  enquire  why  these  particular  methods  of  combining 
the  differential  equations  so  frequently  furnish  the  solution  of  physical 
problems,  we  are  led  to  the  conclusion  that  the  quantities  whose  exact 
differentials  are  introduced  by  the  combination  correspond  to  some- 
thing really  existing  in  nature.  It  is  thus  made  probable  on  purely 
theoretical  grounds  that  kinetic  and  potential  energy  are  not  mere 
abstractions,  but  have  an  objective  reality,  and  that  the  conservation 
of  energy  is  a  law  of  nature. 


56  KINETICS   OF   A   PARTICLE.  [104. 

2.     CENTRAL    FORCES. 

104.  We  proceed  to  apply  the  general  principles  developed 
in  the  preceding  articles  to  the  motion  of  a  particle  under  the 
action  of  central  forces. 

The  term  central  force  is  generally  understood  to  imply  two- 
conditions,  viz.  (a)  that  the  direction  of  the  force  passes  constantly 
through  a  fixed  point,  usually  called  the  centre  of  force  ;  and  (b) 
that  the  magnitude  of  the  force  is  a  function  of  the  distance  from 
the  centre  alone  (comp.  Art.  56). 

Let  O  be  the  centre  of  force,  P  the  position  of  the  moving 
particle  at  any  time  t,  m  the  mass  of  the  particle,  and  OP=r  its 
distance  from  the  centre  ;  then  the  general  expression  for  a 
central  force  Fis 


where  the  function  F(r)  represents  the  law  of  force,  and  the 
function  f(r)  the  law  of  the  acceleration  produced  by  this  force 
in  the  particle  m. 

105.  The  most  important  special  case  is  that  of  a  force  pro- 
portional to  some  power  of  the  distance  r,  say 

where  //,  and  n  are  constants.  The  constant  /*,  which  represents 
the  value  of  the  force  at  unit  distance  from  the  centre,  is  often 
called  the  intensity  of  the  force,  or  of  the  centre. 

In  the  case  of  Newton's  law  of  universal  gravitation  (Part  II., 
Art.  257)  we  have  n=  —  2,  p^icmm',  where  K  is  a  constant,  viz. 
the  acceleration  produced  by  a  unit  of  mass  acting'  on  a  unit  of 
mass  at  unit  distance,  while  m  is  the  mass  of  the  attracted  par- 
ticle, and  m1  that  of  the  attracting  centre ;  that  is,  Newton's 
law  is  expressed  by  the  formula 


106.    From  the  physical  point  of  view,  attractions  following  Newton's 
law,  and  indeed,  central  forces  generally,  are  usually  regarded  as  due  to 


io8.]  CENTRAL   FORCES.  57 

the  presence  of  mass,  not  only  in  the  moving  particle,  but  also  at  the 
centre  of  force ;  and  the  action  between  these  two  masses  is  then  a 
mutual  action,  being  of  the  nature  of  a  stress,  i.e.  consisting  of  two 
equal  and  opposite  forces.  It  follows  that  what  we  have  called  the 
centre  of  force  is  not  a  fixed  point. 

It  will,  however,  be  shown  later  (Arts.  150-157)  that  a  simple  modi- 
fication allows  us  to  apply  to  this  case  the  results  deduced  on  the  assump- 
tion that  the  centre  is  fixed. 

Again,  the  attracting  or  repelling  masses  will  here  be  regarded  as  con- 
centrated at  points.  It  should  be  remembered  that  a  homogeneous 
sphere,  according  to  Newton's  law,  attracts  a  particle  outside  of  its 
mass  as  if  the  whole  mass  of  the  sphere  were  concentrated  at  the 
centre  of  the  sphere  (Part  II.,  Arts.  272-276).  •  The  attraction  of  any 
other  mass  on  a  particle  can,  of  course,  always  be  reduced  to  a  single 
force  ;  but  as  the  particle  moves,  the  direction  of  this  force  will  not 
in  general  pass  through  a  fixed  point ;  such  a  force  is,  therefore,  not 
central. 

107.  If  a  particle  P  of  mass  m  be  acted  upon  by  a  single 
central  force 

F=mf(r), 

its  acceleration  j=F/m=f(r)  will  pass  through  the  centre  of 
force  and  be  a  function  of  r  alone.  The  problem  reduces, 
therefore,  at  once  to  the  kinematical  problem  of  central  motion 
(Part  I.,  Art.  223).  Although  the  leading  ideas  of  the  solution 
of  this  problem  have  been  indicated  in  kinematics  (Part  L,  Arts. 
225-234,  237-238),  the  importance  of  the  subject  of  central 
forces  demands  a  restatement  in  this  place  of  the  principal 
methods  in  the  language  of  kinetics,  and  a  more  complete  expo- 
sition of  some  special  cases. 

108.  A  particle  of  mass  m  acted  upon  by  a   single   central 
force  F=mf(r)  will  describe  a  curvilinear  path  whenever  the 
initial  velocity  is  different  from  zero  and  does  not  pass  through 
the  centre  of  force   (see  Art.    56).     As  shown  in  kinematics 
(Part  L,  Art.  225),  the  path  of  the  particle,  here  usually  called 
the  orbit,  is  always  a  plane  curve. 


5  8  KINETICS   OF   A   PARTICLE.  [109. 

Taking  the  plane  of  motion  as  ;rj/-plane  and  the  centre  O 
as  origin  (Fig.  16),  the  direction  cosines 
of  the  force  F  are  ^x/r,  ~3-y/r,  the 
upper  sign  corresponding  to  an  attrac- 
tive force,  the  lower  to  a  repulsion. 
L_  Hence,  the  dynamical  equations  of 

Fig.  16.  motion  are 


If  mf(r)  be  substituted  for  F,  the  factor  m  disappears,  and 
the  equations  become  purely  kinematical. 

109.  To  avoid  the  use  of  the  double  sign,  we  shall  give  the 
equations  in  the  form  corresponding  to  the  more  important  case 
of  attraction  ;  for  a  repulsive  force  it  will  only  be  necessary  to 
change  throughout  the  sign  of  F  or  f(r).  Thus  the  funda- 
mental equations  of  motion  are  (comp.  Part  I.,  Art.  226): 


If  polar  co-ordinates  r,  6  (Fig.  16),  with  the  centre  as  pole,  be 
used,  the  equations  of  motion  are,  since  the  total  acceleration 
is  along  the  radius  vector  : 

,          •       I  d 


110.  Two  principal  problems  present  themselves  :  (a)  the 
problem  of  finding  the  orbit  for  a  given  law  of  force,  and  (b) 
the  converse  problem  of  determining  the  law  of  force,  i.e.  the 
function  /(r),  when  the  orbit  is  given.  The  solution  of  the  former 
problem  is  effected  by  obtaining  first  integrals  of  the  equations 
of  motion  from  the  principle  of  areas  and  from  the  principle 
of  kinetic  energy,  and  by  combining  these  integrals  so  as  to 
effect  a  second  integration.  Formulae  for  the  solution  of  the 
latter  problem  will  be  found  incidentally. 


112.] 


CENTRAL   FORCES. 


59 


111.   The  second  of  the  polar  equations  (3)  gives  immediately, 
if  c  denote  the  constant  of  integration  : 


dt 


(4) 


and  the  meaning  of  this  equation  is  that  the  sectorial  velocity  is 
constant  and  equal  to  \  c.  The  same  result  can  be  obtained  from 
the  equations  (2)  by  applying  the  principle  of  areas  (see 
Arts.  87,  88). 

To  express  the  constant  c  in  terms  of  the  initial  conditions, 
let  v  denote  the  velocity,  /  the  perpendicular  to  it  from  the 
centre  of  force,  and  ty  the  angle  between  the  radius  vector  r 
and  the  velocity  v,  all  at  the  time  /  (Fig.  17) ;  and  let  the  initial 
values  of  these  quantities,  at  the  time  *=o,  be  distinguished  by 


Fig.  17. 


zero-subscripts.     Then  it  follows  from  the  equation  (4)  that  we 
have  (see  Art.  89  and  Part  I.,  Art.  230) 


=  z       sn 


(5) 


i.e.  the  velocity  is  inversely  proportional  to  its  perpendicular  dis- 
tance from  the  centre,  or,  as  it  is  sometimes  expressed,  the 
moment  of  the  velocity  about  the  centre  of  force  is  constant. 

112.  Another  first  integral  of  the  equations  of  motion  is 
obtained  by  combining  the  equations  (i)  according  to  the  prin- 
ciple of  kinetic  energy  (Art.  71).  This  gives 

mv2)  =  -  Fdr,    or  d(%  v2)  =  -f(r)dry  (6) 


60  KINETICS   OF   A   PARTICLE.  [113, 

whence  i^—v^  —  2j    f(r}dr\  (7). 

i.e.  the  velocity  at  any  distance  r  depends  only  on  this  distance 
(besides  the  initial  radius  vector  and  velocity),  and  is  indepen- 
dent of  the  path  described,  being  the  same  as  if  the  particle 
had  been  projected  with  the  initial  velocity  along  the  straight 
line  joining  the  initial  position  to  the  centre. 

113.  To  perform  the  second  integration  we  have  only  to 
substitute  in  (7)  for  v  its  value  in  terms  of  r  and  /  or  r  and  6. 
Now  the  general  expression  for  the  velocity  in  any  curvilinear 
motion  is  (Part  I.,  Art.  142) 


dt        \dt      \dt      ddJ 

From  these  expressions  one  -of  the  variables  0  and  t  can  be 
eliminated  by  substituting  for  d6/dt  its  value  c/r*  from  (4)  ;  this 
gives 


It  is  often  convenient  to  replace  the  radius  vector  r  by  its- 
reciprocal  u=i/r]  we  then  have 

l~  /  J.  \  9  ~l 

(9) 

114.  The  formulas  (4)  and  (7),  together  with  the  expressions 
(8)  or  (9),  contain  the  complete  solution  of  the  two  principal 
problems  mentioned  in  Art.  no.  Thus,  if  the  law  of  force  be 
given,  the  form  of  the  function  f(r)  is  known,  and  v  can  be 
found  from  (7)  in  function  of  r  or  u ;  substituting  this  value  of 
z;  in  either  (8)  or  (9),  we  have  a  differential  equation  of  the  first 
order  between  r  and  /,  or  between  r  and  0.  The  integration  of 
the  latter  equation  gives  the  integral  equation  of  the  orbit. 

On  the  other  hand,  if  the  equation  of  the  path  be  given,  the 
expressions  (8)  or  (9)  furnish  the  value  of  v2,  which,  substituted 
in  (6),  determines  the  law  of  force  f(f). 


:ii6.]  CENTRAL   FORCES.  6l 

When  the  equation  of  the  orbit  is  known,  i.e.  when  r  is  known 
.as  a  function  of  9  or  vice  versa,  the  time  t  of  the  motion  can  be 
found  by  integrating  equation  (4),  viz. 

dt=-r*>dQ. 
c 

115.  If  the  second  expression  for  v2  in  (9)  be  introduced  into 
the  differential  equation  of  kinetic  energy  (6),  we  find 


•or 

This  will  generally  be  found  the  most  convenient  form  for  find- 
ing the  law  of  force  when  the  polar  equation  of  the  orbit  is 
given.  Again,  when/(r)  is  given,  the  integration  of  this  differ- 
ential equation  of  the  second  order  is  often  more  convenient 
for  finding  the  equation  of  the  orbit  than  the  method  indicated 
in  Art.  114. 

It  may  be  noted  that  the  important  relation  (10)  can  be  de- 
rived directly  from  the  equations  of  motion  (3),  by  eliminating  / 
by  means  of  (4)  and  introducing  u  for  i/r.  We  have 


^_  _  2  2 
dt~          '  d&^ 

If  these  values  be  substituted  into  the  first  of  the  equations  (3), 
the  relation  (10)  will  result. 

116.  When  the  equation  of  the  orbit  can  be  expressed  con- 
veniently in  terms  of  r  and/,  as  is,  for  instance,  the  case  for  the 
conic  sections,  it  is  of  advantage  to  combine  the  equation  of 
kinetic  energy,  </(J^2)  =  —f(r)dr,  directly  with  the  equation 
resulting  from  the  principle  of  areas,  pv  —  c.  This  gives 


dr  ~2      dr   '       f  dr 


62  KINETICS   OF  A  PARTICLE.  [117. 

117.  It  is  easy  to  see  that  the  methods  here  explained  would 
apply  even  to  the  more  general  problem  when  the  force  F,  while 
passing  ahvays  through  a  fixed  centre,  is  not  a  function  of  the 
distance  r  alone,  but  a  function  of  both  co-ordinates  r  and  6.  The 
principal  difference  will  appear  in  the  impossibility  of  perform- 
ing directly  the  integration  indicated  in  (7). 

With  F=mf(r,  0),  the  equation  of  kinetic  energy  is,  for 
attraction, 


and  substituting  for  v2  the  first  of  the  values  given  in  (8),  we 
find 


This  equation  shows  that  the  motion  relative  to  the  radius 
vector  takes  place  as  if  the  actual  resulting  force  F=mf(r,  6} 
were  increased  by  an  additional  force  m<?/r*. 

For  the  law  of  force  we  have,  as  in  Art.  115: 


118.  We  proceed  to  the  consideration  of  some  special  cases. 
The  most  important  of  these  are  the  case  of  a  force  directly 
proportional  to  the  distance,  and  that  of  a  force  inversely  pro- 
portional to  the  square  of  the  distance. 

119,  Force  Proportional  to  the  Distance  :  f(r)  =  K2r.    The  equations 
of  motions  (2)  are  in  this  case 


, 

the  upper  sign  holding  for  attraction,  the  lower  for  repulsion.  Their 
solution  is  very  simple,  because  each  equation  can  be  integrated  sepa- 
rately. We  find,  in  the  case  of  attraction, 

x  =  #!  cos  K.t  4-  a.2  sin  *t,  y  =  b^  cos  K/  -f-  b%  sin  K/, 
and  in  the  case  of  repulsion, 

x  =  alf»  +  a#-Kt,  y  =  V  +  V*  ; 
<*i»  az>  b±,  fit  being  the  constants  of  integration. 


122.]  CENTRAL   FORCES.  63- 

120,  To  find  the  equation  of  the  orbit,  it  is  only  necessary  to  eliminate 
t  in  each  case. 

In  the  case  of  attraction,  this  elimination  can  be  performed  by  solv- 
ing for  cos  Kt,  sin  K/,  squaring  and  adding.     The  result  is 

OiJF  -  ^i*)2  +  (a*y  -  b&Y  =  (a A  -  0 A)2, 
and  this  represents  an  ellipse,  since 

(af  +  a/)  W  +  tt)  ~  ("A  +  a  AY  =  (a  A  ~  <*AY 
is  always  positive.     The  centre  of  the  ellipse  is  at  the  origin,  and  the 
lines  a±y  =  d^x,  a2y  =  bgx  are  a  pair  of  conjugate  diameters. 

121,  In  the  case  of  repulsion,  solve  for  eKt  and  e~Kt,  and  multiply. 
The  resulting  equation, 

(&\y  -  b\x)  (fa  —  <*zy)  =  (a A  —  <*A)2, 
represents   a   hyperbola   whose    asymptotes   are   the    lines    aly  =  blx, 


122.  It  is  worthy  of  notice  that  the  more  general  problem  of  the 
motion  of  a  particle  attracted  by  any  number  of  fixed  centres,  with  forces 
directly  proportional  to  the  distances  from  these  centres,  can  be  reduced 
to  the  problem  of  Art.  119. 

Let  x,  y,  z  be  the  co-ordinates  of  the  particle,  ri  its  distance  from  the 
centre  Oi ;  xit  yit  zt  the  co-ordinates  of  Ot ;  and  —  K(V(  the  acceleration 
produced  by  Ot.  Then  the  .^-component  of  the  resultant  acceleration  is 

and  similar  expressions  obtain  for  the  y  and  z  components.     Hence,  the 
equations  of  motion  are 


As  these  expressions  are  linear  in  x,  y,  z,  there  is  one,  and  only  one, 
point  at  which  the  resultant  acceleration  is  zero.  Denoting  its  co- 
ordinates by  x,  y,  z,  we  have 


'  = 


The  form  of  these  equations  shows  that  this  point  of  zero  acceleration, 
which  is  sometimes  called  the  mean  centre,  is  the  centroid  of  the  centres 


64 


KINETICS   OF   A   PARTICLE. 


[123- 


of  force,  if  these  centres  be  regarded  as  containing  masses  equal  to  K?. 
It  is  evidently  a  fixed  point. 

123.   By  introducing  the  co-ordinates  of  the  mean  centre,  we  can 
now  reduce  the  equations  of  motion  to  the  simple  form 


where  K2  =  SK/.     Finally,  taking  the  mean  centre  as  origin,  we  have 

\--ty, 


u  •*  _  _  K2~ 

*•"      • 


^  =  -A. 

,^u2 


It  thus  appears  that  the  motion  of  the  particle  is  the  same  as  if  there  were 
only  a  single  centre  of  force,  viz.  the  mean  centre  (x,y,z),  attracting 
with  a  force  proportional  to  the  distance  from  this  centre. 

The  plane  of  the  orbit  is,  of  course,  determined  by  the  mean  centre 
and  the  initial  velocity.  The  equation  of  this  plane  can  be  found  by 
applying  the  principle  of  areas  (Art.  94). 

124,    It  is  easy  to  see  that  most  of  the  considerations  of  Art.  122 
apply  even  when  some  or  all  of  the  centres  repel the  particle  with  force 
proportional  to  the  distance.     It  may,  however,  happen  in  this  case  tha 
the  mean  centre  lies  at  infinity,  in  which  case  it  can,  of  course,  not  be 
taken  as  origin. 

Simple  .geometrical   considerations  can  also  be  used  to   solve  the 

problem.     Thus,   in   the   case   of  two 
attractive    centres    O1}    O2    (Fig.    18) 
of  equal  intensity  *2,  the   forces   can 
evidently  be   represented   by  the  dis 
tances  POi  =  rlt  PO2  =  r2  of  the  par 
tide    P    from     the     centres.      Thei 
resultant    is   therefore    =  2  PO,   if   O 
denotes  the  point  midway  between  O 
and    <92;    and    this    resultant    alway 
passes  through  this  fixed  point  O,  and  is  proportional  to  the  distance 
PO  from  this  point. 


Fig.   18. 


125.   Exercises. 

(i)  Determine  the  constants  of  integration  in  Art.  119,  if  x0,y0  are 
the  co-ordinates  of  the  particle  at  the  time  /=o  and  x0,  j0  the  com 


125.]  CENTRAL   FORCES.  65 

ponents  of  its  velocity  v0  at  the  same  time.     The  equation  of  the  orbit 
will  assume  the  form 


for  attraction,  and 

*?(*oy  -jo*)2  -  (xQy  -jo*)2  = 

for  repulsion. 

(2)  Show  that   the   semi-diameter   conjugate   to  the  initial,  radius 
vector  has  the  length  VQ/K,  where  v<?  =  x02  +  j>02.     As  any  point  of  the 
orbit  can  be  regarded  as  initial  point,  it  follows  that  the  velocity  at  any 
point  is  proportional  to  the  parallel  diameter  of  the  orbit. 

(3)  Find  what  the  initial  velocity  must  be  to  make  the  orbit  a  circle 
in  the  case  of  attraction,  and  an  equilateral  hyperbola  in  the  case  of 
repulsion. 

(4)  The  initial  radius  vector  r0  and  the  initial  velocity  z>0  being  given 
geometrically,  show  how  to  construct  the  axes  of  the  orbit  described 
under  the  action  of  a  central  force  (of  given  intensity  K2)  proportional 
to  the  distance  from  the  origin. 

(5)  A  particle  describes  an  ellipse  under  the  action  of  a  central 
force  proportional  to  the  distance  ;    show  that  the  eccentric    angle  is 
proportional   to  the  time,   and  find  the  corresponding   relation   for  a 
hyperbolic  orbit. 

(6)  A  particle  of  mass  m  describes  a  conic  under  the  action  of  a 
central  force   jp=^.  m^r.      Show  that   the  sectorial  velocity  is  \  c  = 

r  Kab,  a  and  b  being  the  semi-axes  of  the  conic. 

(7)  In  Ex.  (6)  show  that  the  time  of  revolution  is  T=  2?r/K,  if  the 
conic  is  an  ellipse. 

(8)  A  particle  describes  a  conic  under  the  action  of  a  force  whose 
direction  passes  through  the  centre  of  the  conic.     Show  that  the  force 

s  proportional  to  the  distance  from  the  centre. 

(9)  A   particle  is  acted  upon  by  two  central  forces  of  the  same 
ntensity  (*2),  each  proportional  to  the  distance  from  a  fixed  centre. 

Determine  the  orbit  :  (a)  when  both  forces  are  attractive  ;  (b)  when 
Doth  are  repulsive;  (c)  when  one  is  an  attraction,  the  other  a  re- 
uulsion. 

PART  in  —  5 


66  KINETICS    OF   A   PARTICLE.  [126. 

(10)  A  particle  of  mass  m  is  attracted  by  two  centres  O1}  O2  of 
equal  mass  m'  and  repelled  by  a  third  centre  (93,  whose  mass  is 
m"  =  2  m'.  If  the  forces  are  all  directly  proportional  to  the  respective 
distances,  determine  and  construct  the  orbit. 

(n)  When  a  particle  moves  in  an  ellipse  under  a  force  directed 
towards  the  centre,  find  the  time  of  moving  from  the  end  of  the  major 
axis  to  a  point  whose  polar  angle  is  0. 

126,   Force  Inversely  Proportional  to  the  Square  of  the  Distance  : 

f(r)={ji/r*  (Newton's  law). 

It  has  been  shown  in  kinematics  (Part  I.,  Arts.  229-236)  how 
this  law  of  acceleration  can  be  deduced  from  Kepler's  laws  of 
planetary  motion.  From  Kepler's  first  law  Newton  concluded 
that  the  acceleration  of  a  planet  (regarded  as  a  point  of  mass 
m)  is  constantly  directed  towards  the  sun  ;  from  the  second  he 
found  that  this  acceleration  is  inversely  proportional  to  the 
square  of  the  distance.  The  motion  of  a  planet  can  therefore 
be  explained  on  the  hypothesis  of  an  attractive  force, 


issuing  from  the  sun. 

The  value  of  /u,  which  represents  the  acceleration  at  unit 
distance  or  the  so-called  intensity  of  the  force,  was  found  to  be 
(Part  I.,  Art.  236;  or  below,  Art.  139) 


and  as,  according  to  Kepler's  third  law,  the  quantity  aB/T2  has 
the  same  value  for  all  the  planets,  Newton  inferred  that  the 
intensity  of  the  attracting  force  is  the  same  for  all  planets  ;  in 
other  words,  that  it  is  one  and  the  same  central  force  that 
keeps  the  different  planets  in  their  orbits. 

127.  It  was  further  shown  by  Newton  and  Halley  that  the 
motions  of  the  comets  are  due  to  the  same  attractive  force. 
The  orbits  of  the  comets  are  generally  ellipses  of  great  eccen- 


128.]  CENTRAL   FORCES.  67 

tricity,  with  the  sun  at  one  of  the  foci.  As  a  comet  is  within 
range  of  observation  only  while  in  that  portion  of  its  path 
which  lies  nearest  to  the  sun,  a  portion  of  a  parabola,  with  the 
same  focus  and  vertex,  can  be  substituted  for  this  portion  of 
the  elliptic  orbit,  at  least  as  a  first  approximation. 

It  is  also  found  from  observation  that  the  motions  of  the 
moons  or  satellites  around  the  planets  follow  very  nearly 
Kepler's  law.  A  planet  can  therefore  be  regarded  as  attracting 
each  of  its  satellites  with  a  force  proportional  to  the  mass  of 
the  satellite  and  inversely  proportional  to  the  distance. 

128.  All  these  facts  led  Newton  to  suspect  that  the  force  of 
terrestrial  gravitation,  as  observed  in  the  case  of  falling  bodies 
on  the  earth's  surface,  might  be  the  same  as  the  force  that 
keeps  the  moon  in  its  orbit  around  the  earth.  This  inference 
could  easily  be  tested,  since  the  acceleration  g  of  falling  bodies 
as  well  as  the  moon's  distance  and  time  of  revolution  were 
known. 

Let  m  be  the  mass  of  the  moon,  a  the  major  semi-axis  of  its  orbit, 
T  the  time  of  revolution,  r  the  distance  between  the  centres  of  earth  and 
moon;  then  the  earth's  attraction  on  the  moon  is  (Art.  126) 


or,  since  the  eccentricity  of  the  moon's  orbit  is  so  small  that  the  orbit 
can  be  regarded  as  nearly  circular, 


On  the  other  hand,  the  attraction  exerted  by  the  earth  on  a  mass  m  on 
its  surface,  i.e.  at  the  distance  R  =  3963  miles  from  the  centre,  must  be 

f"  =  mg. 

Now,  if  these  forces  are  actually  in  the  inverse  ratio  of  the  squares  of 
the  distances,  we  must  have 


68  KINETICS    OF   A   PARTICLE.  [129. 

or,  since  the  distance  of  the  moon  is  nearly  =  60  R, 

F'  =  6o2F. 
Substituting  the  above  values  of  F  and  F\  we  find 

^=47T2X603X^2. 

With  R—  3963  miles,  T=  27*  f  43"*,  this  gives 

£•=32-0, 

a  value  which  agrees  sufficiently  with  the  observed  value  of  g,  consider- 
ing the  rough  degree  of  approximation  used. 

129,  In  this  way  Newton  was  finally  led  to  his  law  of  universal 
gravitation,  which  asserts  that  every  particle  of  mass  m  attracts 
every  other  particle  of  mass  m'  with  a  force 


r2    ' 

where  r  is  the  distance  of  the  particles  and  K  a  constant,  viz. 
the  acceleration  produced  by  a  unit  of  mass  in  a  unit  of  mass  at 
unit  distance  (see  Part  II.,  Art.  257,  261-262). 

The  best  proof  of  this  hypothesis  as  an  actual  law  of  physical 
nature  is  found  in  the  close  agreement  of  the  results  of  theo- 
retical astronomy  based  on  this  law  with  the  observed  celestial 
phenomena. 

It  may  be  noticed  that,  according  to  this  law,  -the  path  of  a 
projectile  in  vacuo  is  only  approximately  parabolic,  the  actual 
path  being  a  very  elongated  ellipse  or  hyperbola,  one  of  whose 
foci  is  at  the  earth's  centre. 

130.  Taking  Newton's  law  as  a  basis,  let  us  now  turn  to  the 
converse  problem  of  determining  the  motion  of  a  particle  acted 
upon  by  a  single  central  force  for  which  f(r)  =  p/r*  (problem 
of  planetary  motion). 

It  has  been  shown  in  kinematics  (Part  I.,  Arts.  239-242)  that 
if  the  force  be  attractive,  the  particle  will  describe  a  conic  section 


I3i.]  CENTRAL   FORCES.  69 

with  one  of  the  foci  at  the  centre  of  force,  the  conic  being  an 
ellipse,  parabola,  or  hyperbola,  according  as 

V|2f  (13) 

>  ro 

If  the  force  be  repulsive,  the  same  reasoning  will  apply,  except 
that  IJL  is  then  a  negative  quantity.  The  orbit  is,  therefore,  in 
this  case  always  hyperbolic ;  the  branch  of  the  hyperbola  that 
forms  the  orbit  must  evidently  turn  its  convex  side  towards  the 
focus  at  which  the  centre  of  force  is  situated,  since  the  force 
always  lies  on  the  concave  side  of  the  path. 

131.  To  exhibit  fully  the  determination  of  the  constants  and  the 
dependence  of  the  nature  of  the  orbit  on  the  initial  conditions,  a  solution 
somewhat  different  from  that  given  in  kinematics  will  here  be  given  for 
the  problem  of  planetary  motion  in  its  simplest  form. 

With/(r)=/x/V2,  the  equation  of  kinetic  energy,  (7),  Art.  112,  gives 


or,  if  the  constant  of  integration  be  denoted  briefly  by  h  and  u=i/rbe 
introduced :     ' 

v*  =  2fjiU  +  h,  where  h  =  v£ — -•  (14) 

ro 

Substituting  this  expression  of  i?  into  the  equation  (9),  Art.  113,  we 
find  the  differential  equation  of  the  orbit  in  the  form 


'2 

or 


To  integrate,  we  introduce  a  new  variable  «'  by  putting 


the  resulting  equation, 


KINETICS   OF   A   PARTICLE. 


has  the  general  integral 
0  —  a  = 


cos 


u',  or  u*  =  cos  (6  —  a), 


where  a  is  the  constant  of  integration.      The  orbit  has,  therefore,  the 
equation 


c1 


(16) 


which  agrees  in  form  with  the  equation  (74)  given  in  kinematics  (Part 
I.,  Art.  242),  excepting  the  different  notation  used  for  the  constants. 

132,  The  equation  (16)  represents  a  conic  section  referred  to  its 
focus  as  origin.  The  general  focal  equation  of  a  conic  is 

lie 

-  =  -7  +  -,  cos  (0—  a),  (17) 

r      I      I 

where  /  is  the  semi-latus  rectum,  or  parameter,  e  the  eccentricity,  and 
a.  the  angle  made  with  the  polar  axis  by  the  line  joining  the  focus  to  the 
nearest  vertex. 

In  a  planetary  orbit  (Fig.  19),  the  sun  S  being  at  one  of  the  foci, 
the  nearest  vertex  A  is  called  the  perihelion,  the  other  vertex  A'  the 
aphelion,  and  the  angle  0  —  a  made  by  any  radius  vector  SP=  r  with 
the  perihelion  distance  SA  is  called  the  true  anomaly. 

Comparing  equations  (17)  and 
(16),  we  find,  for  the  determina- 
tion of  the  constants  : 


Fig.  19. 


hence, 


or,  solving  for  c  and  h, 

<r  =  V/I7,    h  =  p.—j 


(i9) 


133.   The  expression  for  the  eccentricity  e  in  (18)  determines  the 
nature  of  the  conic ;  the  orbit  is  an  ellipse,  parabola,  or  hyperbola, 

according  as  *=i;  hence,  by  (18),  according   as  the  constant  h  of 


136.]  CENTRAL   FORCES.  71 

the  equation  of  kinetic  energy  is  negative,  zero,  or  positive.  Owing  to 
the  value  of  h  given  in  (14),  this  criterion  agrees  with  the  form  (13), 
Art.  130. 

It  should  be  observed  that  it  follows  from  (13)  that  the  nature  of  the 
conic  is  independent  of  the  direction  of  the  initial  velocity. 

134.  The  criterion  (13)  can  be  given  the  following  interpretation. 
Consider  a  particle  attracted  by  a  fixed  centre  according  to  Newton's 
law.  If  it  move  in  a  straight  line  passing  through  the  centre,  the 
principle  of  kinetic  energy  gives  for  its  velocity,  at  the  distance  r, 


hence,  if  it  start  from  rest  at  an  infinite  distance  from  the  centre,  it 
would  acquire  the  velocity  V2/z/r  at  the  distance  r.  The  criterion  (13) 
is  therefore  equivalent  to  saying  that  the  orbit  is  an  ellipse,  a  parabola, 
or  a  hyperbola,  according  as  the  velocity  at  any  point  is  less  than,  equal 
to,  or  greater  than  the  velocity  which  the  particle  would  have  acquired  at 
that  point  by  falling  towards  the  centre  from  infinity  (comp.  Art.  57). 

135.  For  a  central  conic,  whose  axes  are  20,  2b,  we  have  /=  $  /a, 
e  =  V#2  qp  &l  a  (the  upper  sign  relating  to  the  ellipse,  the  lower  to  the 
hyperbola),  so  that  the  equations  (19)  reduce  to  the  following: 

h  =  ^£.  (20) 

a 

The  latter  relation,  with  the  value  of  h  from  (14),  gives  for  the  major 
or  real  semi-axis  a  : 


while  the  former,  with  the  value  of  c  as  given  in  (5),  Art.  in,  deter- 
mines the  minor  or  transverse  axis  b  : 

(22) 

136.  The  magnitudes  of  the  axes  having  thus  been  found,  their 
directions  can  be  determined  by  a  simple  construction  which  furnishes 
the  second  focus. 


KINETICS    OF   A   PARTICLE. 


[137. 


O" 


In  the  ellipse,  the  focal  radii  have  a  constant  sum  =  2<z,  and  lie  on 
the  same  side  of  the  tangent,  making  equal  angles  with  it.     In  the 

hyperbola,  they  have  a  con- 
stant difference  =  20,  and  lie 
on  opposite  sides  of  the  tan- 
gent. 

Hence,  determining  the 
point  <9"  (Fig.  20),  which  is 
symmetrical  to  the  centre  of 
force  O  with  respect  to  the 
initial  velocity,  and  drawing 


O' 


Fig.  20. 


O 


the  line  P0O",  we  have  only  to  lay  off  on  this  line  from  PQ  a  length 
P0O'  =  ±  (2a  —  r0)  ;  then  O'  is  the  second  focus,  which  for  an  elliptic 
orbit  must  be  taken  with  O  on  the  same  side  of  the  tangent  PT,  and  for 
a  hyperbolic  orbit  on  the  opposite  side. 


137.    For  a  parabola,  since  e=  i,  we  find,  from  (19), 


(23) 


The  axis  of  the  parabola  is  readily  found  by  remembering  that  the 
perpendicular  let  fall  from  the  focus  on  the  tangent  bisects  the  tangent 
(i.e.  the  segment  of  the  tangent  between  the 
point  of  contact  and  the  axis).  Hence,  if 
OT  (Fig.  21 )  be  the  perpendicular  let  fall 
from  the  centre  O  on  the  velocity  z>0,  it  is 
only  necessary  to  make  TT'  =  P0T,  and  T' 
will  be  a  point  of  the  axis.  Moreover,  the 
perpendicular  let  fall  from  T  on  OT  will 
meet  the  axis  at  the  vertex  A  of  the  parabola, 
so  that  OA  =  i  /. 


138.   The  relation  (21),  which  must  evi- 


Fig.  21 


dently  hold  at  any  point  of  the  orbit,  can  be  written  in  the  form 


2a 


(24) 


the  upper  sign  relating  to  the  ellipse,  the  lower  to  the  hyperbola,  while 
for  the  parabola,  the  second  term  in  the  parenthesis  vanishes  (since 
a  =  oo). 


140.]  CENTRAL   FORCES. 


73 


This  convenient  expression  for  the  velocity  in  terms  of  the  radius 
vector  might  have  been  derived  directly  from  the  fundamental  relation 
(5),  v  =  c/p,  the  first  of  the  equations  (19),  (?  =  //,/,  and  the  geometrical 
properties  of  the  conic  sections  (r  ±  r'  =  2a,  pp'  =  &2,  p'r  =  pr',  where 
r,  r1  are  the  focal  radii,  and  /,  /'  the  perpendiculars  let  fall  from  the  foci 
on  the  tangent)  .  The  proof  is  left  to  the  student. 

139.  Time.  In  the  case  of  an  elliptic  orbit,  the  time  T  of  a  complete 
revolution,  usually  called  the  periodic  time,  is  found  by  remembering 
that  the  sectorial  velocity  is  constant  and  =  1  c  (Art.  in),  whence 


rp  _ 

or,  by  (20),  T=2          =      .  (25) 

The  constant  n 


which  evidently  represents  the  mean  angular  velocity  in  one  revolution,. 
is  called  the  mean  motion  of  the  planet.  It  should  be  noticed  that  it 
depends  not  only  on  the  intensity  of  the  force,  but  also  on  the  major 
axis  of  the  orbit,  while  in  the  case  of  a  force  directly  proportional  to  the 
distance  it  is  independent  of  the  size  of  the  orbit  (see  Art.  125,  Ex.  7). 
The  periodic  time  7"and  the  major  axis  a  of  a  planetary  orbit  deter- 
mine the  intensity  p.  of  the  force 

^  =  4-2^,  (26)- 

whence  F=  mf(r)  =  m^  =  4*?i*jzp  (27) 

where  m  is  the  mass  of  the  planet. 

140.  To  find  generally  the  time  /  in  terms  of  0  or  r,  we  can,  of  course, 
proceed  as  indicated  in  Art.  114;  but  the  resulting  expressions  are 
somewhat  complicated,  and  it  is  best  to  introduce  the  eccentric  angle 
<£  of  the  ellipse  as  a  new  variable,  and  to  express  /,  r,  and  6  in  terms  of 
<£.  In  astronomy,  the  polar  angle  0  is  known  as  the  true  anomaly,  and 
the  eccentric  angle  <£  as  the  eccentric  anomaly. 


74 


KINETICS   OF   A   PARTICLE. 


[141. 


141.    The  relation  of  the  eccentric  angle  <j>  to  the  polar  co-ordinates 
r,  0  will  appear  from  Fig.  22,  in  which  P  is  the  position  of  the  planet 


Fig.  22. 

at  the  time  /,  P'  the  corresponding  point  on  the  circumscribed  circle, 
^.AOP—  0  the  true  anomaly,  and  %  ACP'=(f>  the  eccentric  anomaly. 
The  focal  equation  of  the  ellipse 

r=_  _L =  «(t-«Y 

i  +  e  cos  6      i  +  e  cos  0 

gives  r  +  er  cos  6 — a — ae1 ;  and  the  figure  shows  that  r  cos  0 = a  cos  <J> — ae 
hence, 

r=.a(\  —  e  cos<£),  or  a  —  r=z  ae  cos  <£.  (28' 

Equating  this  value  of  r  to  that  given  by  the  polar  equation  of  the 
ellipse,  we  have 


i  —  e  cos 


,    orcosO= 


i  +  e  cos  0 '  i  —  e  cos  <f> 

A  more  symmetrical  form  can  be  given  to  this  relation  by  computing 


i-cos0=2sin2-=(i+<?)  I- 


i  —  e  cos  <j> 

i  +  cos  0  =  2  cos2-  =  (i  —  e)   T  +  cos  9 
2  i  —  e  cos  <J> 


whence,  by  division,         tan-  =  \/^— t-^ 

2       V  !_^ 


(29) 


142.  To  find  t  in  terms  of  r,  we  have  only  to  substitute  in  (24)  for 
iP  its  value  from  (8),  Art.  113,  and  to  integrate  the  resulting  differential 
equation 


I43-]  CENTRAL   FORCES.  75 

As,  by  (20),  Art.  135,  ci  =  ^/a  =  ^a(i  —  e2),  this  equation  becomes 


or 


la  rdr 

dt=  \/  -- 

VVaV-o- 


The  integration  is   easily  performed   by  introducing   the   eccentric 
angle  <J>  as  variable  by  means  of  (28)  ;  this  gives 


=  \- 


•  a(i  —  e  cos 


If  the  time  be  counted  from  the  perihelion  passage  of  the  planet,  we 
have  /=  o  when  r  =  a  —  ae,  i.e.  when  $  =  o  ;  hence,  putting  V/*/tf3=  n, 
.as  in  Art.  139,  we  find 

nt  =  <f>  —  e  sin  <£.  (30) 

This  relation  is  known  as  Kepler's  equation  ;  the  quantity  nt  is  called 
the  mean  anomaly. 

143.  Kepler's  equation  (30)  can  be  derived  directly  by  considering 
that  the  ellipse  APA  (Fig.  22)  can  be  regarded  as  the  projection  of 
the  circle  AP'A',  after  turning  this  circle  about  A  A  '  through  an  angle 
=  cos"1  (£/#).  For  it  follows  that  the  elliptic  sector  A  OP  is  to  the 
•circular  sector  A  OP'  as  b  is  to  a  .  Now,  for  the  circular  sector  we  have 

A  OP  =  A  CP  -  O  CP'  =  J  flfy  -  1  ae  -  a  sin  <£  =  -  (<£  -  e  sin  <£)  ; 
hence,  the  elliptic  sector  described  in  the  time  /  is 
AOP=-  •  ' 


a  2 

The  sectorial  velocity  being  constant  by  Kepler's  first  law,  we  have 


hence,  /=  —  (<£- 

27T 

and  this  agrees  with  (30)  since,  by  (25),  27r/7T=  «. 


76  KINETICS    OF   A   PARTICLE.  [144. 

144.  Kepler's  equation  (30)  gives  the  time  as  a  function  of  <j>  ;  by 
means  of  (28),  it  establishes  the  relation  between  t  and  r\  by  means 
of  (29),  it  connects  /  with  0.  It  is,  however,  a  transcendental  equation 
and  cannot  be  solved  for  <£  in  a  finite  form. 

For  orbits  with  a  small  eccentricity  e,  an  approximate  solution  can 
be  obtained  by  writing  the  equation  in  the  form 

<£  =  nt-\-  e  sin<£, 
and  substituting  under  the  sine  for  <£  its  approximate  value  nt  : 

<f>  =  nt  +  e  sin  nt.  (31) 

This  amounts  to  neglecting  terms  containing  powers  of  e  above  the 
first  power. 

Substituting  this  value  of  <£  in  (28),  we  have  with  the  same  approxi- 
mation 

r=  a(i  —  ecosnt).  (32) 

To  find  0  in  terms  of  /,  we  have,  from  the  equation  of  the  ellipse, 
r  —  a  (  i  —  e2)  (  i  -f-  e  cos  6)  ~l  =  a  (  i  —  e  cos  0)  ,  neglecting  again  terms  in 
&  ;  hence,  rz  =  02(i  —  2e  cos  0)  .  Substituting  this  value  in  the  equation 
of  areas,  rzdO  =  cdt—  V/xtf  (i  —  e*)dt,  we  find 


(i  -  2e  cos  0)dO  =  Y^<#  =  ndt; 
whence,  by  integration,  since  &  —  o  for  /  =  o, 

0  —  2e  sin  0  =  nt, 
or  finally,  6  =  nt  +  2e  sin  nt.       .  (33) 

Thus  we  have  in  (31),  (32),  (33)  approximate  expressions  for  <f>, 
r,  and  6  directly  in  terms  of  the  time.  The  quantity  2<?  sin  «/,  by  which 
the  true  anomaly  0  exceeds  the  mean  anomaly  nt,  is  called  the  equation 
of  the  centre. 

145,   Exercises. 

(1)  A  particle  describes   an   ellipse   under  the  action  of  a  central 
force.     Determine  the  law  of  force  by  means   of  formula  (n),  Art. 
116:   (a)  when  the  centre  of  force  is  at  the  centre  of  the   ellipse; 
(b)  when  it  is  at  a  focus. 

(2)  A  particle  is  attracted  by  a  fixed  centre  according  to  Newton's 
law.     What  must  be  the  initial  velocity  if  the  orbit  is  to  be  circular  ? 


146.]  CENTRAL   FORCES.  77 

(3)  A  number  of  particles  are  projected,  from  the  same  point  in 
the  field  of  a  force  following  Newton's  law,  with  the  same  velocity,  but 
in  different  directions.     Show  that  the  periodic  times  are  the  same  for 
all  the  particles. 

(4)  The  mean  distance  of  Mars  from  the  sun  being  1.5237  times 
that  of  the  earth,  what  is  the  time  of  revolution  of  Mars  about  the  sun  ? 

(5)  A  particle  describes  a  conic  under  the  action  of  a  central  force 
following  Newton's  law;    if  the  intensity  /x  of  the  force  be   suddenly 
changed  to  /*',  what  is  the  effect  on  the  orbit? 

(6)  In  Ex.  (5),  if  the  original  orbit  was  a  parabola  and  the  intensity 
be  doubled,  what  is  the  new  orbit  ? 

(7)  Regarding  the  moon's  orbit  about  the  earth  as  circular,  what 
would   it  become  :    (a)  if  the  earth's  mass  were   suddenly   doubled  ? 
(£)  if  it  were  reduced  to  one-half  ? 

(8)  In  Ex.  (5),  determine  the  effect  on  the  major  semi-axis  (or 
"mean  distance")  a  and  on  the  periodic  time  T,  of  a  small  change 
in  the  intensity  /A  of  the  force. 

(9)  If  the   mass  of  the   sun  be   suddenly  increased  by  a  small 
amount  while  the  earth  is  at  the  end  of  the  minor  axis  of  its  orbit, 
what  would  be  the  effect  on  the  earth's  mean  distance  and  on  the 
period  of  revolution  T  ? 

(10)  Find  the  equation  of  the  hodograph  of  planetary  motion, 
derive  from  it  the  expression  for  the  velocity  in  terms  of  the  radius 
vector,  and  show  that  the  velocity  is  a  maximum  in  perihelion  and 
a  minimum  in  aphelion. 

(n)  Show  that  the  greatest  velocity  of  a  planet  in  its  orbit  about 
the  sun  is  to  its  least  velocity  as  Vi  +e  is  to  Vi  —  e ;  and  find  this 
ratio  for  the  earth,  whose  orbit  has  the  eccentricity  <?  =  0.01677120. 

(12)   Find  the  time  exactly  as  a  function  of  0,  for  a  parabolic  orbit. 

146.  Force  any  Function  of  the  Distance.  The  general  methods 
have  been  given  in  Arts.  108-1 16.  The  equation  of  energy, 
<6),  Art.  112,  gives,  with  u=i/r, 

u}du     i  /     \ 

r-+*5  (34) 


;8  KINETICS   OF  A   PARTICLE.  [147. 

hence,  substituting  for  v  its  value  from  (9),  Art.  113,  we  find, 
for  the  differential  equation  of  the  orbit, 


As  it  is  often  difficult  or  impossible  to  perform  the  integra- 
tion in  finite  form,  it  is  of  importance  to  determine  the  apses 
and  apsidal  distances  of  the  orbit. 

147.  An  apse  is  a  point  of  the  orbit  at  which  the  velocity  is 
at  right  angles  to  the  radius  vector  drawn  from  the  centre  of 
force  ;  the  length  of  the  radius  vector  of  an  apse  is  called  the 
apsidal  distance,  and  its  direction  an  apsidal  line. 

The  importance  of  the  apsidal  lines  lies  in  the  fact  that  they 
are  lines  of  symmetry  of  the  orbit,  while  the  apsidal  distances 
are  maximum  or  minimum  values  of  the  radius  vector.  This 
will  be  seen  from  the  following  considerations. 

By  the  above  formula  (34)  the  velocity  is  a  function  of  the 
radius  vector  alone  ;  and  by  (5),  Art.  in,  since  sini|r=£/W,  the 
angle  -fy  between  radius  vector  and  velocity  is  also  a  function  of 
the  radius  vector  alone.  It  follows  that,  if  the  velocity  be 
reversed  in  direction  at  any  point  of  the  orbit,  the  same  orbit 
will  be  described  in  the  opposite  sense  ;  and  as  at  an  apse  the 
velocity  is  perpendicular  to  the  apsidal  line,  the  two  portions  of 
the  orbit  on  opposite  sides  of  an  apsidal  line  must  be  symmet- 
rical with  respect  to  this  line. 

148.  The  condition  for  an  apse  is  therefore 


Substituting  this  value  in  the  above  equation  (35),  the  apsidal 
distances  i/u  can  be  found  by  solving  the  equation  for  u.  The 
value  of  du/dO  should  also  change  sign  as  the  particle  passes 
through  the  apse. 


1 5o.]  PROBLEM    OF   TWO   BODIES.  79, 

If  the  law  of  force  is  given  as  a  single-valued  function  f(u), 
there  can  exist  only  two  different  apsidal  distances  (although 
there  may  be  any  number  of  apses).  The  angle  between 
these  two  different  apsidal  distances  is  called  the  apsidal  angle. 

149.   Exercises, 

(1)  Find   the   law   of    force   when   the    equation   of   the   orbit  is 
rn  =  a  cos  nO  +  b,  and  investigate  the  particular  cases  n  —  —  i,  n  =  —  2, 

n  =  i,  n  =  2. 

(2)  A  particle  moves  in  a  circle  under  the  action  of  a  single  force  of 
constant  direction ;  determine  the  law  of  force  and  discuss  the  motion. 

(3)  Find  the  law  of  the  central  force  directed  to  the  origin  under 
whose  action  a   particle   will   describe    the  following  curves :   (a)   the 
spiral  of  Archimedes  r—aO;   (b}  the  hyperbolic  spiral  6r=  a -}   (c}  the 
logarithmic  or  equiangular  spiral  r=aeaQ;   (d)  the  curve  r=acosn&. 

(4)  A  particle  moves  in  a  circle  under  the  action  of  a  central  force 
directed  towards  a  point  on  the  circumference ;  find  the  law  of  force. 

(5)  A  particle  is  acted  upon  by  a  force  perpendicular  to  a  given 
plane  and  inversely  proportional  to  the  cube  of  the  distance  from  the 
plane.     Determine  its  motion. 

(6)  A  particle  moves  in  a  semi-ellipse  under  the  action  of  a  force 
perpendicular  to  the  axis  joining  the  ends  of  the  semi-ellipse.     Deter- 
mine the  law  of  force  and  the  velocity  at  the  ends. 


3.     THE    PROBLEM    OF   TWO    BODIES. 

150.  In  the  preceding  discussion  of  the  motion  of  a  particle 
under  the  action  of,  a  central  force,  it  has  been  assumed  that 
the  centre  of  force  is  fixed.  In  the  applications  of  the  theory 
of  central  forces  this  assumption  is  in  general  not  satisfied. 
Thus,  in  considering  the  motion  of  a  planet  around  the  sun, 
the  force  of  attraction  is,  according  to  Newton's  law  of  universal 
gravitation  (Art.  129),  regarded  as  due  to  the  presence  of  a 
mass  M  at  the  centre  (sun),  and  of  a  mass  m  at  the  attracted 
point  (planet) ;  and  the  action  between  these  two  masses  is  a 


.So  KINETICS   OF   A   PARTICLE.  [151. 

mutual  action,  being  of  the  nature  of  a  stress  •,   i.e.   consisting 
of  two  equal  and  opposite  forces,  each  equal  to 


Hence,  the  mass  m  of  the  planet  attracts  the  mass  M  of  the 
sun  with  precisely  the  same  force  with  which  the  mass  M 
of  the  sun  attracts  the  mass  m  of  the  planet.  The  attraction 
affects,  therefore,  the  motions  of  both  bodies. 

151.  The  accelerations  produced  by  the  two  forces    are,   of 
course,    not    equal.        Indeed,    the    acceleration   F/m  =  rcM/r2, 
produced  in  the  planet  by  the  sun,  is  very  much  greater  than 
the  acceleration  F/M=Km/r2y  produced    by  the  planet  in  the 
sun  ;  for  the  mass  of  even  the  largest  planet  (Jupiter)  is  less 
than    one   thousandth  of   that  of   the  sun.      The  assumption 
of  a  fixed  centre  can  therefore  be  regarded  as  a  first  approxima- 
tion in  the  problem  of  the  motion  of  a  planet  about  the  sun. 

In  the  case  of  the  earth  and  moon,  the  difference  of  the 
masses  is  not  so  great,  the  mass  of  the  moon  being  nearly 
one  eightieth  of  that  of  the  earth. 

It  can,  however,  be  shown  that  the  results  deduced  on  the 
assumption  of  a  fixed  centre  can,  by  a  simple  modification 
be  made  available  for  the  solution  of  the  general  problem  of  tJu 
"motions  of  two  particles  of  masses,  m,  M,  subject  to  no  forces 
besides  their  mutual  attraction.  In  astronomy,  this  is  callec 
the  problem  of  two  bodies.  In  the  solution  below  we  assume  the 
attraction  to  follow  Newton's  law  of  the  inverse  square  o: 
the  distance.  It  will  be  convenient  to  speak  of  the  two 
particles,  or  bodies,  as  planet  (m)  and  sun  (M). 

152.  With  regard  to  any  fixed  system  of  rectangular  axes 
let  x,  y,  z  be  the  co-ordinates  of  the  planet  (m),  at  the  time  t 
x',  y',  z1  those  of  the  sun  (M),  at  the  same  time  ;   so  that  for 
their  distance  r  we  have 


153.]  PROBLEM    OF   TWO    BODIES.  8l 

Then  the  equations  of  motion  of  the  planet  are 

Mm     x'-x 


Mm      v'  — 


t  r  r 

while  the  equations  of  motion  of  the  sun  are 

mM     x—x* 


n 

M—=-  =K 
dt* 


_     mM     z- 
"      ~ 


153.   By  adding  the  corresponding  equations  of  the  two  sets, 
we  find 


If  it  be  remembered  that  the  centroid  of  the  two  masses  m,  M 
has  the  co-ordinates 


_  _ 

' 


m+M  m  +  M  m  +  M 

it  appears  that  these  equations  can  be  written  in  the  form 
d*x 


in  words  :  the  acceleration  of  the  common  centroid  of  planet  and 
sun  is  zero  ;  i.e.  this  centroid  moves  with  constant  velocity  in 
a  straight  line. 

It  may  be  noticed  that  this  result  is  merely  a  special  case 
of  a  more  general  proposition  to  be  proved  hereafter,  viz. 
that  the  centroid  of  any  system  acted  upon  by  no  forces 
external  to  the  system  moves  uniformly  in  a  straight  line 
(Art.  381). 

PART   III  —  6 


82  KINETICS   OF   A   PARTICLE.  [154. 

154.  The  integration  of  the  equations  (i)  would  give  the 
absolute  path  of  the  planet.  But  the  constants  could  not  be 
determined,  because  the  absolute  initial  position  and  velocity 
of  the  planet  are,  of  course,  not  known.  The  same  holds  for 
the  absolute  path  of  the  sun.  All  we  can  do  is  to  determine 
the  relative  motion,  and  we  proceed  to  find  the  motion  of  the 
planet  relative  to  the  sun. 

Taking  the  sun's  centre  as  new  origin  for  parallel  axes,  we 
have  for  the  co-ordinates  f,  77,  f  of  the  planet  in  this  new 
system, 

Now,  dividing  the  equations  (i)  by  m,  the  equations  (2)  by 
M,  and  subtracting  the  equations  of  set  (2)  from  the  corre- 
sponding equations  of  set  (i),  we  find  for  the  relative  accelera- 
tions of  the  planet 

</2f_        M+m    £ 

7^> —       ^          9       *  ~> 

dfi  r*        r 

^        I'  <3> 

M+m    ? 

—  f£  — •          • 


The  form  of  these  equations  shows  that  the  relative  motion  of 
the  planet  with  respect  to  the  sun  is  the  same  as  if  the  sun  were 
fixed  and  contained  the  mass  M  +  m.  Thus  the  problem  is 
reduced  to  that  of  a  fixed  centre,  the  only  modification  being 
that  the  mass  of  the  centre  M  should  be  increased  by  that  of 
the  attracted  particle  m. 

155.  This  result  can  also  be  obtained  by  the  following  simple  con- 
sideration. 'The  relative  motion  of  the  planet  with  respect  to  the  sun 
would  obviously  not  be  altered  if  geometrically  equal  accelerations  were 
applied  to  both.  Let  us,  therefore,  subject  each  body  to  an  additional 
acceleration  equal  and  opposite  to  the  actual  acceleration  of  the  sun 
(whose  components  are  obtained  by  dividing  the  equations  (2)  by  M). 


, 


158.]  PROBLEM   OF   TWO   BODIES.  83 

Then  the  sun  will  be  reduced  to  equilibrium,  while  the  resulting  accel- 
eration of  the  planet,  which  is  its  relative  acceleration  with  respect  to 
the  sun,  will  evidently  be  the  sum  of  the  acceleration  exerted  on  it  by 
the  sun,  and  the  acceleration  exerted  on  the  sun  by  the  planet.  This  is 
just  the  result  expressed  by  the  equations  (3). 

156.  It  can  here  only  be  mentioned  in    passing  that,  while 
the  problem  of  two  bodies  thus  leads  to  equations   that   can 
easily  be  integrated,  the  problem  of  three  bodies  is  one  of  exceed- 
ing difficulty,  and  has  been  solved  only  in  a  few  very  special 
cases.      Much  less  has  it  been    possible  to   integrate  the   3/2 
equations  of  the  problem  of  n  bodies. 

157.  According  to  the  equations  (3),  the  first  and  second  laws 
of  Kepler  can  be  said  to  hold  for  the  relative  motion  of  a  planet 
about  the  sun  (or  of  a  satellite  about  its  primary).     The  third 
law  of  Kepler  requires  some  modification,  since  the  intensity  of 
the  centre  p  should  not  be  =/cM,  but  =fc(M+m).     Thus  we 
have,  by  (26),  Art.  139, 


in  other  words,  the  quotient  a?/T2  is  not  independent  of  the 
mass  m  of  the  planet. 

Thus,  if  mlt  m2  be  the  masses  of  two  planets,  av  a2  the  major 
semi-axes  of  their  orbits,  and  7\,  T2  their  periodic  times,  we 
have 


This  quotient  is  approximately  equal  to  one  if  M  is  very  large 
in  comparison  with  both  m^  and  m^  ;  hence,  for  the  orbits  of  the 
planets  about  the  sun,  Kepler's  law  is  very  nearly  true. 

158.   Exercises. 

(  i  )  Two  particles  of  masses  mlf  m2  attract  each  other  with  a  force 
which  is  any  function  of  the  distance  r  between  them,  say  ^=m1m2/(r). 
Show  that  their  common  centroid  moves  uniformly  in  a  straight  line, 
and  find  the  equations  of  this  line. 


84  KINETICS   OF   A   PARTICLE.  [159. 

(2)  In  Ex.  (i),  write  out  the  equations  for  the  relative  motion  of 
either  particle  with  respect  to  the  common  centroid. 

159.  The  theory  of  central  forces  is  treated  with  considerable  elabo- 
ration in  most  works  on  theoretical  mechanics ;  a  few  references  only 
will  here  be  given  :  P.  APPELL,  Traite  de  mecanique  rationnelle ,  Paris, 
Gauthier-Villars,  1893,  Vol.  I.,  pp.  354-405  ;  B.  WILLIAMSON  and  F.  A. 
TARLETON,  An  elementary  treatise  on  dynamics,  London,  Longmans 
(New  York,  Appleton),  2d  edition,  1889,  pp.  147-205  ;  P.  G.  TAIT  and 
W.  J.  STEELE,  A  treatise  on  dynamics  of  a  particle,  London  and  New 
York,  Macmillan,  6th  edition,  1889,  pp.  113-166;  W.  H.  BESANT,  A 
treatise  on  dynamics,  Cambridge,  Bell  (New  York,  Macmillan),  2d 
edition,  1893,  pp.  120-166,  and  267-275  ;  W.  WALTON,  A  collection  of 
problems  in  illustration  of  the  principles  of  theoretical  mechanics,  Cam- 
bridge, Bell,  3d  edition,  1876,  pp.  248-297.  All  these  works  contain 
numerous  examples  for  practice.  The  theory  of  planetary  motion  is,  of 
course,  treated  in  works  on  theoretical  astronomy.  The  student  will 
also  consult  with  advantage  :  W.  SCHELL,  Theorie  der  Bewegung  und  der 
Kr'dfte,  Leipzig,  Teubner,  2d  edition,  Vol.  I.,  1879,  pp.  373-387; 
E.  BUDDE,  Allgemeine  Mechanik  der  Punkte  und  starren  Systeme, 
Berlin,  Reimer,  Vol.  I.,  1890,  pp.  170-181;  B.  PRICE,  A  treatise  on 
analytical  mechanics,  Oxford,  Clarendon  Press  (New  York,  Macmillan), 
Vol.  I.  (=  Vol.  III.  of  A  treatise  on  infinitesimal  calculus},  2d  edition, 
1868,  pp.  508-574;  O.  RAUSENBERGER,  Lehrbuch  der  analytischen 
Mechanik,  Leipzig,  Teubner,  Vol.  I.,  1888,  pp.  32-102,  where  the 
problem  of  planetary  motion  is  very  fully  discussed;  T.  DESPEYROUS, 
Cours  de  mecanique,  avec  des  notes  par  G.  Darboux,  Paris,  Hermann, 
1884,  Vol.  I.,  pp.  336-369,  427-440,  and  Vol.  II.,  pp.  38-57,  461-466 ; 
and  others. 


i6i.]  CONSTRAINED   MOTION.  85 

• 

IV.     Constrained  Motion. 

I.     INTRODUCTION. 

160.  It  has  been  shown,  in  the  preceding  sections,  that  the 
motion  of  a  free  particle  is  fully  determined  if  all  the  forces 
acting  upon  the   particle,  as  well  as  the  so-called  initial  con- 
ditions, are  given.     The   motion   of   a   particle  may,  however, 
depend  not  only  on  given  forces,  but  on  other  conditions  not 
directly  expressed   in   terms   of  forces.     The   motion  is  then 
said  to  be  constrained. 

Some  of  the  more  important  forms  of  constraint  have  been 
considered  in  Part  II.,  Arts.  218-225.  To  mention  some  more 
concrete  examples :  a  heavy  particle  sliding  down  a  smooth 
inclined  plane  is  subject  not  only  to  the  force  of  gravity,  but 
also  to  the  condition  that  it  cannot  pass  through  the  plane  ;  a 
railway  train  running  on  the  rails,  a  piece  of  machinery  slid- 
ing in  a  groove  or  between  guides,  can,  for  many  purposes,  be 
regarded  as  a  particle  constrained  to  a  curve;  the  bob  of  a 
pendulum,  a  stone  attached  to  a  cord  and  swung  around  by 
the  hand,  may  be  regarded  as  constrained  to  a  surface. 

161.  Sometimes  these  constraining  conditions  can  be  easily 
replaced   by  forces.     Thus,  in  the  first  illustration  above,  the 
condition  that  the  particle   cannot   pass   through  the  inclined 
plane  can  be   expressed   by   introducing   the   reaction  of  the 
plane,  i.e.    a   force    acting   on  the  particle  at  right   angles  to 
the  plane,  so  as  to  prevent  it  from  passing  through  the  plane. 
Similarly,  in  the  case  of   the  stone  attached  to  the  cord,  we 
may  imagine  the  cord  cut  and  its  tension  introduced  so  as  to 
replace  the  condition  by  a  force. 

Whenever  the  constraints  to  which  a  particle  is  subjected 
can  thus  be  expressed  by  means  of  forces,  these  forces  can  be 
combined  with  the  other  impressed  forces,  and  then,  of  course, 
the  equations  of  motion  for  a  free  particle  can  be  applied. 


86  KINETICS   OF   A   PARTICLE.  [162. 

Thus,  let  X',  Yf,  Zf  be  the  components  of  the  resultant  of  all 
the  constraints  ;  X,  Y,  Z  those  of  the  resultant  of  all  the  other 
impressed  forces.  Then  the  equations  of  motion  are  : 


162.  It  must,  however,  be  noticed  that  the  reactions  repre- 
senting the  constraints,  such  as  the  tension  of  the  string  in 
the  example  referred  to,  are   generally  not   given  beforehand. 
Moreover,  the  constraints  are  often  expressed  more  conveniently 
by  conditional  equations.     Thus,  if  the  motion  of  a  particle  be 
restricted  to  a  surface,  the  equation  of  this  surface,  say 

<!>{?,  ytz)=o,  (2) 

may  be  given  as  a  constraining  condition  to  be  fulfilled  by  the 
co-ordinates  of  the  moving  particle. 

163.  As  a  particle  has  but  three  degrees  of  freedom,  it  can 
be  subjected  to  only  one  or  two  conditions  of  the  form  (2). 
One  such  condition  confines  it  to  a  surface  ;  two  to  the  curve 
of   intersection   of   the  two  surfaces    represented  by  the  two 
conditional  equations  ;  three  conditions  would  evidently  prevent 
it  entirely  from  moving. 

164.  The  curve  or  surface  to  which  a  particle  is  constrained 
may  vary  its  position  and  even  its  shape  in  the  course  of  time. 
In  this  case  the  conditional  equations,  referred  to  fixed  axes, 
will  contain  not  only  the  co-ordinates,  but  also  the  time.     That 
is,  they  will  be  of  the  more  general  form 

4>(x,y,z,t)=o.  (3) 

165.  To  constrain  a  particle  completely  to  a  surface,  we  may 
imagine  it  confined  between  two  infinitely  near  impenetrable 
surfaces.     The  complete  constraint  to  a  curve  might  be  realized 
by  confining  the  particle  to  an  infinitely  narrow  tube  having 
the  shape  of  the  curve,  or  by  regarding  it  as  a  ring  sliding  along 
a  wire. 


167.]  MOTION   ON   A   FIXED   CURVE.  87- 

In  many  cases,  however,  the  constraint  is  not  complete,  but 
•only  partial,  or  one-sided.  Thus,  the  rails  compelling  the  train 
to  move  in  a  definite  curve  do  not  prevent  its  being  lifted  verti- 
cally out  of  this  curve,  nor  does  the  cord  that  confines  the 
motion  of  the  stone  to  a  sphere  prevent  it  from  moving  towards 
the  inside  of  the  spherical  surface. 

While  complete  constraints  are  generally  expressed  by  equa- 
tions, one-sided  constraints  should  properly  be  expressed  by 
inequalities.  Thus,  in  the  case  of  the  stone,  the  condition  is 
really  that  its  distance  r  from  the  hand  is  not  greater  than  the 
length  /  of  the  cord,  i.e. 


but  as  soon  as  r  becomes  less  than  /,  the  constraining  action 
-ceases,  and  the  stone  becomes  free.  It  is,  therefore,  in  general 
:sufficient  to  consider  conditional  equations  ;  but  the  nature  of 
the  constraint,  whether  complete  or  partial,  must  be  taken  into 
account  to  determine  when  and  where  the  constraint  ceases  to 
•exist. 

166.  We  now  proceed  to  consider  separately  the  motion  of 
a  particle  constrained  to  a  fixed  curve  and  that  of  a  particle 
•constrained  to  a  fixed  surface.     After  these  special  cases,  the 
general  problem  of  motion  on  a  movable  curve  or  surface  will 
-be  discussed. 

/ 

2.     MOTION    ON    A    FIXED    CURVE. 

167.  The  condition  that  a  particle  should  move  on  a  given 
fixed  curve  can  always  be  replaced  by  introducing  a  single  addi- 
tional force  F'  called  the  constraining  force,  or  the  constraint. 
An  example  will  best  show  how  this  force  can  be  determined. 

Let  us  consider  a  particle  of  mass  m,  subject  to  the  force  of 
gravity  F=mg  alone;  in  general  it  will  describe  a  parabola 
whose  equation  can  be  found  if  the  initial  conditions  are  known. 
To  compel  the  particle  to  describe  some  other  curve,  say  a  verti- 


88 


KINETICS   OF   A   PARTICLE. 


[168. 


cal  circle,  a  constraining  force  F'  (Fig.  23)  must  be  intro- 

duced such  that  the  resultant 
R  of  F  and  F'  shall  produce 
the  acceleration  required  for 
motion  in  the  circle.  Thus,  for 
instance,  for  uniform  motion 
in  a  circle  the  resulting  ac- 
celeration must  be  directed 
towards  the  centre  and  must 
be  =a>2#,  if  a  is  the  radius  and 
a)  the  constant  angular  veloc- 
ity. We  have,  therefore,  in 
this  case  R  —  mo^a  along  the 
radius,  F=  mg  vertically  down- 

wards ;  and  hence,  denoting  by  9  the  angle  made  by  the  radius 

CP  with  the  vertical  (Fig.  23), 

F'2  =  F2  +  R2  +  2  FR  cos  6 

a  cos  6). 


The  constraint  F',  which  is  thus  seen  to  vary  with  the  angle 
6,  can  be  resolved  into  a  tangential  component  Ft  and  a  normal 
component  Fn'.  As  in  our  problem  the  velocity  is  to  remain  of 
constant  magnitude,  the  tangential  constraint  must  just  counter- 
balance the  tangential  component  Ft  =  mgsin  0  of  gravity.  The 
normal  constraint  FJ  not  only  counterbalances  the  normal  com- 
ponent Fn=mgcos  0  of  gravity,  but  also  furnishes  the  centrip- 
etal force  R  =  mco2a  required  for  motion  in  the  circle  ;  i.e. 

0). 


168.  In  the  general  case  of  a  particle  of  mass  m  acted  upon 
by  any  given  forces  and  constrained  to  any  fixed  curve,  it  is 
convenient  to  resolve  both  the  resultant  Fof.  the  given  forces 
and  the  constraint  F1  along  the  tangent  and  the  normal  plane. 


I79-]  MOTION   ON   A   FIXED   CURVE.  3^ 

The  equations  of  motion  (see  Art.  67)  can  then  be  written  in 
the  form 

m  —  =  Pt-Ft\ 

o 

m  —  =  resultant  of  Fn  and  Fn', 
P 

where  v  is  the  velocity  and  p  the  radius  of  curvature  of  the 
path  at  the  time  t.  It  should  be  noticed  that  the  components 
Fn  and  Fn't  though  both  situated  in  the  normal  plane,  do  not 
in  general  have  the  same  direction.  But  in  the  important 
special  case  of  plane  motion,  i.e.  when  the  path  is  a  plane  curve 
and  the  resultant  F  of  the  given  forces  lies  in  this  plane,  Fn  and 
F^  are  both  directed  along  the  radius  of  curvature  so  that  the 
right-hand  member  of  the  second  equation  becomes  the  sum  or 
difference  of  Fn  and  Fn. 

169.  The  normal  component  Fn'  of  the  constraining  force 
is  generally  denoted  by  the  letter  N  and  is  called  the  resistance 
or  reaction  of  the  curve;  a  force  —  N,  equal  and  opposite  to 
this  reaction,  represents  the  pressure  exerted  by  the  particle 
on  the  curve. 

The  tangential  component  Ft'  of  the  constraint  will  exist  only 
when  the  constraining  curve  is  rough,  i.e.  offers  frictional  resist- 
ance ;  we  have  then,  denoting  the  coefficient  of  friction  by  //,, 


We  shall  therefore  write  the  equations  of  motion  as  follows  : 


V 

m  —  =  resultant  of  Fn  and  N.  (2) 

170.  The  normal  component,  mv^/p,  of  the  effective  force 
is  sometimes  called  the  centripetal  force  (see  Art.  67)  ;  it  is 
directed  along  the  principal  normal  of  the  path  towards  the 


g0  KINETICS    OF   A   PARTICLE.  [171. 

centre  of  curvature.  A  force  equal  and  opposite  to  this  cen- 
tripetal force,  i.e.  =  —miP/r,  is  called  centrifugal  force.  It 
should  be  noticed  that  this  is  a  force  exerted  not  on  the 
moving  particle,  but  by  it. 

It  appears  from  equation  (2)  that  the  normal  reaction  N  is 
the  resultant  of  the  centripetal  force  mv^/p  and  the  reversed 
normal  component  of.  the  given  forces,  —  Fn.  Changing  all  the 
signs,  we  can  express  the  same  thing  by  saying  that  the  pressure 
on  the  curve,  —  N,  is  the  resultant  of  the  centrifugal  force,  —  mv2/^, 
and  of  the  normal  component  Fn  of  the  given  forces. 

If,  in  particular,  this  normal  component  Fn  is  zero,  the  press- 
ure on  the  curve  is  equal  to  the  centrifugal  force.  This  case 
is  of  frequent  occurrence.  Thus,  if  a  small  stone  attached  to 
a  cord  be  whirled  around  rapidly,  the  action  of  gravity  on  the 
stone  can  be  neglected  in  comparison  with  the  centripetal  force 
due  to  rotation  ;  hence  the  centrifugal  force  measures  approxi- 
mately the  tension  of  the  string,  and  may  cause  it  to  break. 
Again,  when  a  railway  train  runs  in  a  curve,  the  centrifugal 
force  produces  the  horizontal  pressure  on  the  rails,  which  tends 
to  displace  and  deform  the  rails. 

171.  It  may  happen  that  at  a  certain  time  t  the  pressure  —N 
vanishes.  If  the  constraint  be  complete  (Art.  165),  this  would 
merely  indicate  that  the  pressure  in  passing  through  zero. 
inverts  its  sense.  If,  however,  the  constraint  be  one-sided,  the 
consequence  will  be  that  the  particle  at  this  time  leaves  the 
constraining  curve  ;  for  at  the  next  moment  the  pressure  will 
be  exerted  in  a  direction  in  which  the  particle  is  free  to  move. 

Now  Evanishes  when  its  components  —  Fn  and  witf/p  become 
equal  and  opposite.  The  conditions  under  which  the  particle 
would  leave  the  curve  are,  therefore,  that  the  resultant  Fot  the 
given  forces  should  lie  in  the  osculating  plane  of  the  path,  and 
that  F=z 


172.    To  obtain  the  equations  of  motion  expressed  in  rec- 
tangular Cartesian  co-ordinates,  let  X,  Y,  Z  be  the  components  of 


173-]  MOTION   ON   A   FIXED   CURVE.  91 

the  resultant  F  of  the  given  forces,  and  7VZ,  Ny,  Ng  those  of  the 
normal  reaction  TV  of  the  curve.  If  there  be  friction,  the  fric- 
tional  resistance  pN,  being  directed  along  the  tangent  to  the 
path  opposite  to  the  sense  of  the  motion,  has  the  direction 
cosines  —  dx/ds,  —  dy/ds,  —  dz/ds,  so  that  the  components  of  the 
force  of  friction  are  —^Ndx/ds,  —  ^Ndy/ds^  —pNdz/ds.  The 
general  equations  of  motion  are,  therefore, 


If  the  acceleration  of  the  particle  be  zero,  the  left-hand  mem- 
bers are  all  =o,  and  the  equations  reduce  to  the  conditions  of 
equilibrium  of  a  particle  on  a  curve,  as  given  in  Statics  (Part 

ii.,  P.  138,  (i4». 

In  addition  to  the  equations  (3)  we  have  of  course  the  equa- 
tions of  the  curve,  say 

<£  (*,  y,  *)  =o,  ^r  (*,  y,  *)=o,  (4) 

and  the  relations 

(5) 


the  latter  expressing  that  TV  is  perpendicular  to  the  element  ds 
of  the  path. 

173.    Multiplying  the  equations  (3)  by  dx,  dy,  dz,  and  adding, 
we  find  the  equation  of  kinetic  energy 

d(%mi?)  =  Xdx+  Ydy+Zdz-^Nds.  (7) 

This  relation  might  have  been  obtained  directly  from  the  con- 
sideration that  for  a  displacement  ds  along  the  fixed  curve  the 
normal  reaction  N  does  no  work,  while  the  work  of  friction  is 


92  KINETICS   OF  A   PARTICLE.  [174. 

174.   Exercises. 

(1)  Show  that  when  the  given  forces  are  zero  and  there  is  no  friction, 
the  particle  moves  uniformly  on  the  curve,  and  the  pressure  on  the  curve 
is  proportional  to  the  curvature  of  the  path. 

(2)  A  particle  of  mass  m  moves  down  a  straight  line  inclined  to  the 
horizon  at  an  angle  0,  under  the  action  of  gravity  alone. 

(a)  If  there- be  no  friction,  we  have  by  Art.  169,  since  p=oo   (see 
Fig.  24). 

dv  .    n 

m  —  =  mg  sin  0, 
dt 

o  =  mg  cos  Q  —  N. 

The  first  of  these  equations  is  the 
same  as  that  derived  in  kinematics 
for  motion  down  an  inclined  plane 
(see  Part  L,  Arts.  164-166).  The 
second  equation  gives  the  normal 
reaction  of  the  line  JV=  mg  cos  6 ; 
hence,  the  pressure  on  the  line, 
—N,  is  constant. 

(b)  If  the  line  be  rough,  the  second  equation  remains  the  same,  while 
the  first  must  be  replaced  by  the  following, 

m  —  =  mg  sin  9  —  pN=  mg(sm  0— /A  cos  6). 
dt 

As  the  acceleration  is  constant  whether  there  be  friction  or  not,  the 

motion  is  uniformly  accelerated,  unless  sin  0  —  /A  cos  9  =  o,  i.e.  /A  =  tan  6. 

Find  v  and  s ;  show  that,  in  the  exceptional  case  /x  =  tan  0,  the  motion 

is  uniform  unless  the  initial  velocity  be  zero ;  show  that,  for  motion  up 

the  plane,  the  first  equation  becomes  dv/dt=.  —  £"(sin  0  +  //.cos0),  the 

motion  being  uniformly  retarded  until  /=  v0/g(sm  0  -\-  /A  cos  0)  when  the 

,  particle  either  begins  to  move  down  the  line  or  remains  at  rest. 

(3)  A  string  of  length  /  (ft.)  carries  at  one  end  a  mass  of  m  Ibs. 
while  the  other  end  is  fixed  at  a  point  O  on  a  smooth  horizontal  table. 
The  mass  m  is  made  to  describe  a  circle  of  radius  /  about  O  on  the 
table,  with  constant  velocity  =  v  ft.  per  second.     Show  that  the  tension 
of  the  string  is  =  mi? /I  poundals. 


I74-]  MOTION    ON   A   FIXED   CURVE.  93 

(4)  In  Ex.  (3),  let  m  =  2  Ibs. ;  /=  3  ft. ;  find  the  tension  in  pounds  : 
(a)  when  the  mass  makes  one  revolution  per  second ;    (t>)   when   it 
makes  10  revolutions  per  second.     (<:)  If  the  "string  cannot  stand  a  ten- 
sion of  more  than  300  pounds,  what  is  the  greatest  allowable  velocity? 

(5)  A  locomotive  weighing  32  tons   moves  in  a   curve   of  800   ft. 
radius  with  a  velocity  of  30  miles  an  hour ;  find  the  horizontal  pressure 
on  the  rails. 

(6)  To  prevent  the  lateral  pressure  on  the  rails  in  a  curve,  the  track 
is  inclined  inwards.     Determine  the  required  elevation  e  of  the  outer 
above  the  inner  rail  for  a  given  velocity  v  and  radius  R  if  the  gauge 
(i.e.  the  distance  between  the  rails)  is  4  ft.  8  in. 

(7)  A  plummet  is  suspended  from  the  roof  of  a  railroad  car;   how 
much  will  it  be  deflected  from  the  vertical  when  the  train  is  running 
45  miles  an  hour  in  a  curve  of  300  yards'  radius? 

(8)  A  body  on  the  surface  of  the  earth  partakes  of  the  earth's  daily 
rotation  on  its  axis.     The  constraint  holding  it  in  its  circular  path  is  due 
to  the  attractive  force  of  the  earth.     Taking  the  earth's  equatorial  radius 
as  3963  miles,  show  that  the  centripetal  acceleration  of  a  particle  at  the 
equator  is  about  -^  ft.  per  second,  or  about  ^^  of  the  actually  observed 
acceleration  g  =  32-09  of  a  body  falling  in  vacua. 

(9)  If  the  earth  were  at  rest,  what  would  be  the  acceleration  of  a 
body  falling  in  vacuo  at  the  equator  ? 

(10)  Show  that  if  the  velocity  of  the  earth's  rotation  were  over 
17  times  as  large  as  it  actually  is,  the  force  of  gravity  would  not  be 
sufficient  to  detain  a  body  near  the  surface  at  the  equator. 

(n)  Show  that  in  latitude  <£  the  acceleration  of  a  falling  body,  if 
the  earth  were  at  rest,  would  be  gi  ==g  -\-j  cos2  <£,  where  g  is  the  observed 
acceleration  of  a  falling  body  on  the  rotating  earth  and  j  the  centripetal 
acceleration  at  the  equator.  Thus,  in  latitude  <£  '=  45°,  g=  980-6  cm. ; 
hence  #  =  982. 3. 

(12)  A  chandelier  weighing  75  Ibs.  is  suspended  from  the  ceiling 
of  a  hall  by  means  of  a  chain  12^-  ft.  long  whose  weight  is  neglected. 
By  how  much  is  the  tension  of  the  chain  increased  if  it  be  set  swinging 
so  that  the  velocity  at  the  lowest  point  is  5  ft.  per  second  ? 

(13)  A  cord  of  2  ft.  length  passes  at  its  middle  point  through  a  hole 
in  a  smooth  horizontal  table.     It  carries  at  its  lower  end  a  mass  of 
2  Ibs.,  at  its  other  end  a  mass  of  i  Ib.     The  latter  is  set  to  revolve  in  a 


94 


KINETICS  OF   A   PARTICLE. 


['75- 


circle  about  the  hole  so  as  to  stretch  the  cord  and  just  prevent  the 
mass  of  2  Ibs.  from  descending,  (a)  How  many  revolutions  must  it 
make?  (b)  If  only  one-fourth  of  the  cord  lie  on  the  table  while  three- 
fourths  hang  down,  how  many  revolutions  must  be  made  ? 

(14)  Show  that,  when  a  particle  moves  with  constant  velocity  in  a 
vertical  circle,  the  constraining  force  F*  (Art.  167)  is  always  directed 
towards  a  fixed  point  on  the  vertical  diameter. 

175.  A  particle  of  mass  m  subject  to  gravity  alone  is  con- 
strained to  move  in  a  vertical  circle  of  radius  1.  If  there  be 
no  friction  on  the  curve  and  the  constraint  be  produced  by  a 
weightless  rod  or  string  joining  the  particle  to  the  centre  of 
the  circle,  we  have  the  problem  of  the  simple  mathematical 
pendulum. 

Equation  (i),  Art.  169,  is  readily  seen  to  reduce  in  this  cas 
(see  Fig.  25)  to  the  form 


: 


(8) 


A  first  integration  gives,  as  shown  in  kinematics  (Part  I.,  Arts. 
215,  216), 


o 


-/cos  00), 


(9) 


where  VQ  is  the  velocity  which  the  particle  has  at  the  time  /=o 
B  when  its  radius  makes  the  angle 

AOP0  =  6Q     with     the     vertical 

M  S  !R  NSV~ -      Multiplying  by  m,  we  have,  for  the 

kinetic  energy  of  the  particle, 


where  h  =  v£/2g—lco$6Q  is 
constant.  If  the  horizontal  line 
MNt  drawn  at  the  height  v^/ 
above  the  initial  point  PQ,  inter 
sect  the  vertical  diameter  AB  at 
R,  it  appears  from  the  figure  that 


178.]  MOTION    ON   A   FIXED    CURVE.  95 

176.  Taking  R  as  origin  and  the  axis  of  z  vertically  down- 
wards, we  have  RQ=z=lcos6+k\  hence  the  force-function  U~ 
has  the  simple  expression 

U—mgz\ 

and  the  velocity  v=^/2gz  is    seen  to  become   zero  when  the 
particle  reaches  the  horizontal  line  MN, 

For  the  further  treatment  of  the  problem,  three  cases  must 
be  distinguished  according  as  this  line  of  zero-velocity  'MN 
intersects  the  circle,  touches  it,  or  does  not  meet  it  at  all  ;  i.e. 
according  as 

A  =  /for—  =2/sin2-.  (11) 

>  2g  >  2 

177.  Equation  (2),  Art.  169,  serves  to  determine  the  reaction 
jVof  the  circle,  or  the  pressure  —TV  on  the  circle.     We  have 


V* 

m  —  =  —mg(. 
N=m(^+i 

;os0  +  ^V; 

f  COS  ^J- 

whence 

Substituting  for  v*  its  value  from  (10),  we  find 

(12). 


The  pressure  on  the  curve  has  therefore  its  greatest  value  when 
6  =  0,  i.e.  at  the'  lowest  point  A.  It  becomes  zero  for  /cos  0X 
=  —  J//,  which  is  easily  constructed. 

178.  If  the  constraint  be  complete  as  for  a  bead  sliding  along 
a  circular  wire,  or  a  small  ball  moving  within  a  tube,  the  press- 
ure merely  changes  sign  at  the  point  0  =  0r  But  if  the  con- 
straint be  one-sided,  the  particle  may  at  this  point  leave  the 
circle.  The  one-sided  constraint  maybe  such  that  OP^  /,  as 
when  the  particle  runs  in  a  groove  cut  on  the  inside  of  a  ring, 
or  when  it  is  joined  to  the  centre  by  a  string  ;  in  this  case  the 


,96  KINETICS   OF   A   PARTICLE.  [179. 

particle  may  leave  the  circle  at  some  point  of  its  upper  half. 
Again,  the  one-sided  constraint  may  be  such  that  OP>1,  as 
when  the  particle  runs  in  a  groove  cut  on  the  rim  of  a  disc ; 
in  this  case  the  particle  can  of  course  only  move  on  the  upper 
half  of  the  circle. 

179.   Exercises. 

(1)  For   h  =  l,  equation   (10)   can  be   integrated   in   finite    terms. 
Show  that  in  this  limiting  case  the  particle  approaches  the  highest  point 
B  of  the  circle  asymptotically,  reaching  it  only  in  an  infinite  time. 

(2)  Derive  the  equations  of  motion  for  the  problem  of  the  simple 
pendulum  (Art.  175)  from  the  general  equations  of  Arts.  172,  173. 

(3)  For  00  ==  60°,  /=  i  ft.,  270  =  9  ft.  per  second,  show  that  the  par- 
ticle will  leave  the  circle  very  nearly  at  the  point  $1=  120°,  if  the  con- 
straint be  such  that  OP<1  (Art.  178). 

(4)  For  z'o=  10  ft.  per  second,  everything  else  being  as  in  Ex.  (3), 
show  that  the   particle  will  leave  the  circle  at  the  point  ^  =  134!°, 
nearly. 

(5)  A  particle,  subject  to   gravity   and   constrained   to   the   inside 
of  a  vertical  circle  (OP^_/),  makes  complete  revolutions.     Show  that 
it  cannot  leave  the  circle  at  any  point,  if  \h  >  /;  and  that  it  will  leave 
the  circle  at  the  point  for  which  cos  0  =  -  f  h/l,  if  f  h  <  I. 

(6)  In  the  experiment  of  swinging  in  a  vertical  circle  a  glass  contain- 
ing water,  and  suspended  by  means  of  a  string,  if  the  string  be  2  ft.  long, 
what  must  be  the  velocity  at  the  lowest  point  if  the  experiment  is  to 
succeed? 

(7)  A  particle  subject  to  gravity  moves  on  the  outside  of  a  vertical 
circle;  determine  where  it  will  leave  the  circle:   (a)  if MN  (Fig.  25)  ' 
intersects  the  circle ;   (b)  if  MN  touches  the  circle ;   (c)  if  MN  does 
not  meet  the  circle. 

(8)  A  particle  subject  to  gravity  is  compelled  to  move  on  any  vertir 
cal  curve  z  =/(#)  without  friction.    Show  that  the  velocity  at  any  point 
is  v  =  ^/2gz  (comp.  Art.  176)  if  the  horizontal  axis  of  x  be  taken  at 
height  above  the  initial  point  equal  to  the  "  height  due  to  the  initi; 
velocity,"  i.c.  v£/2g. 


i8o.]  MOTION    ON    A   FIXED   CURVE.  97 

(9)  Investigate  the  motion  of  a  particle  subject  to  gravity,  and  com- 
pelled to  move  on  a  circle  whose  plane  is  inclined  to  the  horizon  at  an 
angle  a. 

(10)  A  particle  constrained  to  a  straight  line  is  attracted  to  a  fixed 
centre  outside  this  line,  the  attraction  being  proportional  to  the  distance 
from  the  centre.  Determine  its  motion. 

180.  Motion  on  Any  Fixed  Curve  without  Friction.  The  posi- 
tion of  a  point  on  a  curve  can  always  be  determined  by  a  single 
variable.  Thus,  for  instance,  the  length  s  of  the  curve  counted 
from  some  origin  on  the  curve  might  be  taken  as  this  variable  ; 
if  the  curve  be  a  circle,  the  polar  angle  6  might  be  selected  ; 
•on  an  ellipse,  tb^  eccentric  angle  </>  ;  on  a  cycloid,  the  angle 
through  which  the  generating  circle  has  rolled,  etc.  We  shall 
designate  this  variable  by  q,  and  write  the  equations  of  the 
•curve  in  the  form 

*=/i(?).  y=f*(q\  *=/8(?)-  (13) 

The  expression  for  the  velocity  v  is  in  this  case 

/^y 

UJ 

If  there  is  no  friction,  the  real  equation  of  motion  is  the 
•equation  (i)  of  Art.  169,  which  is  equivalent  to  the  equation  of 
kinetic  energy  (7),  Art.  173  ;  when  the  variable  q  is  introduced, 
this  equation  becomes 


where  v2  is  given  by  (14). 
Putting,  for  shortness, 


we  can  write  the  equation  of  motion  in  the  simple  form 

(17) 


.PART   III  —  7 


98  KINETICS   OF  A  PARTICLE.  [181. 

181.  In  the  most  general  case,  the  given  forces  Xy  Y,  Z  will 
depend  not  only  on  the  position  of  the  particle,  but  also  on  its 
velocity  and  on  the  time.     In  this  case,  Q  would  be  a  function 
of  q,  dq/dt,  and  t\  and  equation  (17)  represents  a  differential 
equation  of  the  second  order  between  q  and  /. 

If,  however,  the  resultant  Fof.  the  given  forces  depends  only 
on  the  position  of  the  particle  so  that  Q  is  a  function  of  q 
alone,  the  right-hand  member  of  (17)  is  an  exact  differential, 
and  a  first  integration  can  at  once  be  performed.  Then,  substi- 
tuting for  ?/2  its  value  from  (14)  in  terms  of  q  and  dq/dt,  we  find 
a  differential  equation  of  the  first  order  whose  integration  gives 
/  in  function  of  q. 

182.  Exercise. 

A  particle  of  mass  m  is  constrained  to  a  common  helix  x  =  a  cos  0, 
y  =  a  sin  0,  z  =  *0,  whose  axis  is  vertical.  The  particle  is  subject  to 
gravity  and  is  attracted  by  a  centre  situated  on  the  axis,  with  a  force 
directly  proportional  to  the  distance.  Determine  the  motion. 


3.     MOTION    ON   A   FIXED    SURFACE. 

183.  Just  as  for  motion  on  a  curve  (Art.  172),  we  find  the 
general  equations  of  motion 


dP  '    r    ds* 


The  normal  reaction 

N=^/N*+N?+N*  (2) 

being  at  right  angles  to  the  constraining  surface 


1 84.] 


MOTION   ON  A  FIXED   SURFACE. 


99 


the  following  condition  must  be  satisfied : 

f=f=f<  "         <4) 

fV       9y       9* 

where   </>x,   <f>v,   <f>e  denote,   as   usual,  the  partial  derivatives  of 

x,  yt  z)  with  regard  to  x,  y,  z,  respectively.1 

If  the  acceleration  of  the  particle  be  zero,  the  equations  (i) 
reduce  to  the  conditions  of  equilibrium  of  a  particle  on  a  sur- 
face, as  given  in  Statics  (Part  II.,  Art.  222). 

184.  A  particle  of  mass  m,  subject  to  gravity,  is  constrained  to 
remain  on  the  surface  of  a  sphere  of  radius  r.  If  the  constraint 
is  produced  by  a  weightless  rod  or  string  joining  the  particle 
to  the  centre  of  the  sphere,  the  rod  or  string  describes  a  cone, 
and  the  apparatus  is  called  a  conical  or  spherical  pendulum. 

Taking  the  centre  O  of  the  sphere  as  origin  (Fig.  26),  and 
the   axis   of    z   vertically    down- 
wards, we  have  for  the  equation 
of  the  sphere 

whence  <f>Jx=  $y/y  —  <$>z/z.  The 
direction  cosines  of  ./Vare  —x/r, 
—y/r,  —z/r.  Hence,  the  equa- 
tions of  motion,  as  there  is  no 
friction : 


Fig.  26. 


my—  — 


mz  —  mg 


-N*- 
r 


(6) 


1  This  abridged  notation  is  readily  extended  to  the  second  and  higher  derivatives : 

<t>xx  =  2-2,    <pxy  =   d  ft  f  etc>     it  wni  aiso  sometimes  be  convenient  to  use  the  fluxional 

dx  dxdy 

notation  for  derivatives  with  respect  to  the  time 


dx .      dy .       dz .       d*x . . 

~dt~X'     ~dt~y^     ~dt~Z'       ~di*~*' 


dt* 


dr 


<tt. 
dt 


In  mechanics,  this  notation  is  of  particular  advantage,  not  only  because  the  time 
so  often  appears  as  the  independent  variable,  but  also  because  the  initial  values  of 
these  derivatives  (i.e.  the  components  of  the  initial  velocity  and  acceleration)  can 
then  be  indicated  by  zero  subscripts.  Thus,  the  components  of  the  initial  velocity 
would  be  ^o>  jo>  20- 


100  KINETICS   OF   A   PARTICLE.  [185. 

As   the   resistance   N  does  no   work,  the  principle  of  kinetic 
energy  gives 


or,  dividing  by  \  m, 

V*  =  2(gZ  +  h).  (7) 

To  determine  the  constant  of  integration  hy  we  have  v  —  v^  when 
z—z^  ;  hence 

zs).  (8) 


185.  Another  first  integral  is  found  by  applying  the  principle 
of  areas  which  holds  for  the  projection  of  the  motion  on  the 
horizontal  ^j/-plane.     This   appears  by  considering  that  N  is 
always   directed  along  the  radius  of   the  sphere  so  that  the 
resultant  of  N  and  the  weight  mg  of  the  particle  always  inter 
sects  the  axis  of  z  (see  Art.  93).     We  have  therefore 

where  \c  is  the  sectorial  velocity  of  the  projection  OP'  of  the 
radius  OP=ron  the  .rj/-plane. 

186.  For  the  further  treatment  of  the  problem  it  is  best  to  introduce 
polar  co-ordinates  (Fig.  26).     Let  &  be  the  angle  between  r  and  the 
axis  of  z,  <f>  that  between  the  projection  OP  of  r  on  the  xy-p\a.ne  anc 
the  axis  of  x ;  then 

x=  rsin6cos<f>,    y  =  r  sin  0  sin  <J>,     z  — 
and  x=rcosOcos  <f>>0  —  r  sin  0  sin  <(>•<)>, 

y=r  cos  0  sin  <f>>0  +  r  sin  0  cos  <£•<£, 


Hence  ^  =  j*  +/  +  ?  =  r\fr  +  sin2  0.<£2), 

xy—yx=r*  sin2  &-<j>. 

The  first  integrals  (7)  and  (9)  thus  become  in  polar  co-ordinates 

(10) 
(uj; 


i88.]  MOTION    ON  A  FIXED    SURFACE.  IOi 

187.  The  constants  of  integration  h,  c  can  now  be  determined  if  the 
initial  values  of  0,  0,  <j>  are  given. 

Eliminating  dt  between  the  equations  (10)  and  (n),  we  find  the 
differential  equation  of  the  path 

,,  cdB 

—,  (12) 


sin  B  V2  r2  sin2  0(gr  co$0  +  h)-<? 

whose  integration  gives  the  equation  of  the  path  in  the  spherical  co- 
ordinates 6,  (f>  (colatitude  and  longitude). 

On  the  other  hand,  if  <j>  be  eliminated  between  (10)  and  (n),  we 
find  the  relation 

d       _  r*  sin  0  dQ  _  ,     v 

"  V2  r2  sin20(£rcos  0  +  h)-  / 

which,  upon  integration,  determines  the  time  as  a  function  of  0,  or  the 
position  of  the  point  in  its  path  at  any  given  time. 

The  equations  (12)  and  (13)  contain  therefore  the  complete  solution 
of  our  problem,  with  the  exception  of  the  determination  of  the  resist- 
ance N.  Their  discussion  cannot  here  be  given,  as  they  lead  to  elliptic 
integrals. 

188.  To  find  the  resistance  Nt  multiply  the  equations  (6)  by  x,  yt  z 
and  add  ;  this  gives 

m  (xx  +  yy  +  zz)  =  mgz  —  Nr.  ;  ;  (14) 

Differentiating  twice  the  equation  of  the  sphere  (5),  we  find 

x'x  +yy  +  zz  +  x2  +y  +  2  =  0, 

or  since  x2  +/  +  z2  =  z/2, 

xx  -\-yy  +  zz  =  —  v2. 
Substituting  this  value  in  (14),  we  find 


or,  by  (8),  N=-($gz  +  v<?-  2gz0).  (15) 


If  the  constraint  be  one-sided  as  in  the  case  of  a  string  pendu- 
lum, the  particle  will  leave  the  sphere  whenever  in  its  upper  half  z 
becomes  ^  f  z0  —  v<?/$g. 


102 


KINETICS    OF   A   PARTICLE. 


[189. 


189.   That  particular  case  of  the  problem  of  the  conical  pendulum  in 
which  the  particle  moves  in  a  horizontal  circle  can  be  treated  directly 
in  an  elementary  manner.     It  finds  its  application  in 
the  theory  of  the  governor  of  a  steam  engine. 

Let  O  (Fig.  27)  be  the  point  of  suspension,  OP— 
the  length  of  the  pendulum  rod,  2f.  QOP=  6  the  con- 
stant angle  of  inclination  of  the  rod  to  the  vertical  OQ 
The  only  forces  acting  on  the  particle  are  its  weight  mg 
and  the  tension  of  the  rod.  As  both  these  forces  li< 
in  the  normal  plane  of  the  path,  the  tangential  accel- 
eration is  zero,  and  the  particle  moves  uniformly  in  the 
circle. 

The  radius  of  curvature  of  the  path  is  the  radius 
mflf  QP—  /sin  9  of  the  horizontal  circle.  The  resultant  R 
of  mg  and  N  must  act  along  the  radius  ;  its  magnitude 
is  seen  from  the  figure  to  befi  =  mgtanO.  Hence  th^  equation  (2) 
of  Art.  169  gives  v 


Q 


R\ 


Fig.  27. 


7'2 


/sin0 


=  mgtan.  0, 


or, 


The  figure  also  shows  that  the  tension  of  the  rod  is 


(16) 


COS0 


190.   Exercises. 

(1)  Show  that  the  time  of  revolution   T  of  the  conical  pendulum 
(Art.  1 86)  is  the  same  as  the  time  of  one  complete  oscillation  of  a 
simple  pendulum  of  length  /cos  0. 

(2)  Show  that  the  angular  velocity  with  which  the  vertical  plane 
of  the  rod  turns  about   the  vertical  axis   OQ   (Fig.   27)    is  inversely! 
proportional  to  the  cosine  of  the  angle  0. 

(3)  A  conical    pendulum    makes   n  =  60   revolutions    per    minute : 
(a)  What  is  the  height  of  the  cone  ?     (£)   If  the  mass  of  the  bob  bej 
m—\  oz.,  and  the  length  of  the  rod  /  =  i   ft.,  what  is  the  tension 
of  the  rod?     (^=32-2.) 


I9i.]  MOTION   ON   A  FIXED   SURFACE.  IC>3 

(4)  From  the  equations  (5)  and  (6),  Art.  184,  derive  the  approxi- 
mate path  of  the  bob  of  a  simple  pendulum  when  the  oscillations  are 
very  small. 

191.  Motion  on  Any  Fixed  Surface  without  Friction.  The  posi- 
tion of  a  point  on  a  surface  can  always  be  determined  by  two 
variables,  say  qlt  q^.  Thus,  on  a  sphere,  the  latitude  and  longi- 
tude of  a  point  determine  its  position  ;  and  on  any  surface  the 
two  systems  of  curves  known  as  the  curves  of  curvature  of 
the  surface  might  serve  as  a  system  of  co-ordinates.  In  other 
words,  the  motion  of  a  point  on  a  surface  is  really  a  problem 
of  motion  in  two  dimensions,  just  as  the  motion  on  a  curve 
takes  place  in  one  dimension  (Art.  180). 

Let  *=fi(ffi,  92)>y=A(4i>  ft).  *=/8(ft»  ft)  (l8) 

be  the  equations  of  the  given  surface,  so  that  the  elimination  of 
q-^y  q<i  from  these  equations  would  give  the  ordinary  equation 
<£  (x,  y,  z)=o  of  the  surface.  Then  we  have  for  the  velocity 
v  the  expression 


Kf 


If  there  be  no  friction,  the  equation  of  kinetic  energy  gives 


or  say 

(20) 


If  the  forces  depend  only  on  the  position  of  the  particle,  Ql 
and  <22  are  functions  of  qlt  q^  alone  ;  if,  moreover,  the  expres- 
sion Qidq^+Q^dq^  is  an  exact  differential  of  a  function  of  q^  and 
#2,  say  dU(q^  q%),  the  equation  (20)  gives  at  once  a  first  integral 

q^)+h.  (21) 


104  KINETICS   OF  A  PARTICLE.  [192. 

4.     MOTION    ON    A    MOVING    OR   VARIABLE    CURVE    OR    SURFACE. 

192,  If  the  constraining  curve  or  surface  be  not  fixed  and 
invariable,   the   conditional    equations   will   contain   the  time  t 
explicitly,  besides  the  co-ordinates  xt  y,  z  of  the  moving  particle 
(Art.   164).     For  the  sake  of   simplicity  we  here  assume  the 
curve  or  surface  to  be  smooth,  so  as  to  offer  only  a  normal 
resistance   N\    if    there   be   friction,    the   components   of   the 
frictional  resistance  may  be  regarded  as  included  in  the  com- 
ponents Jf,  Y,  Z  of  the  resultant  force  acting  on  the  particle. 

The  treatment  of  this  general  problem  of  constrained  motion 
of  a  particle  is  here  presented  not  so  much  on  account  of  its 
application  to  the  solution  of  particular  problems,  as  for  the 
reason  that  it  offers  an  opportunity  of  explaining  the  meaning 
of  d'Alembert's  principle  and  illustrating  its  application  in  a 
comparatively  simple  case. 

193.  Two   Constraints.      Let  the   equations  of  the  curve  to 
which  the  particle  is  constrained  be 

<£  (x,  yt  z,  i)  =o,     ^  (x,  y,  z,  t)  =0.  (i) 

To  apply  d'Alembert's  principle  (Arts.  97-102),  let  the  particle 
be  subjected,  at  any  given  time  /,  to  an  infinitesimal  displace- 
ment Bs.  If  this  displacement  be  selected  along  the  curve  (i), 
the  reaction  N  of  the  curve,  being  at  right  angles  to  Bs,  will 
do  no  work  during  the  displacement  ;  hence  the  equation  of 
motion  will  be  the  same  as  that  for  a  free  particle  (see 
Art.  101),  viz. 

(-mx+X)%x+(-my  +  Y)fy  +  (  —  mz+Z)§z=o.          (2) 


In  this  equation,  then,  the  forces  X,  Y,  Z  do  not  involve  the 
normal  reaction  of  the  curve  ;  but  the  components  Bx,  By, 
of  the  displacement  Bs  must  be  selected  so  that  Bs  should  lie  on 
the  curve  (i)  at  the  time  /;  this  is  usually  expressed  by  saying 
that  the  displacement  should  be  compatible  with  the  conditions  (\\ 


I94-]  VARIABLE    CONSTRAINTS.  IO5 

Some  authors  confine  the  term  virtual  displacement  to  dis 
placements  compatible  with  the  given  conditions. 


The  displacement  &$•=  ^/8x2  +  fy/2  +  Sz2  will  be  compatible 
with  the  conditions  (i)  at  the  given  time  t  if  the  following 
conditions,  obtained  by  differentiating  the  equations  (i),  are 
satisfied  : 

$£>x  -f-  fatyr  +  <f)g§z  =  o, 
^  a&tr  +  ^fiy  +  ^fe  =  0.  ^ 

It  should  be  noticed  that  in  this  differentiation  the  time  t  is 
regarded  as  constant,  the  displacement  being  taken  to  occur  at 
a  given  time. 

The  equations  (2)  and  (3),  which  must  be  fulfilled  simulta- 
neously, constitute  the  equations  of  motion  of  our  problem. 

By  means  of  the  equations  (3),  two  of  the  component  dis- 
placements &r,  Sy,  Sz  can  be  eliminated  from  the  equation  (2)  ; 
the  coefficient  of  the  third  equated  to  zero  gives  the  actual 
equation  of  motion. 

194,  To  perform  this  elimination  systematically  the  method  of 
indeterminate  multipliers  can  be  used  as  follows.  Multiplying 
the  equations  (3)  respectively  by  the  indeterminate  factors  X 
and  p  and  adding  them  to  the  equation  (2),  we  obtain  the  single 
equation 


in  which  the  arbitrary  quantities  X,  p  can  be  so  selected  as  to 
make  the  coefficients  of  two  of  the  three  displacements  8x,  fry, 
vanish  ;  the  coefficient  of  the  third  must  then  also  vanish, 
The  equation  is  therefore  equivalent  to  the  following  three 
equations  : 


(4) 
viz  =  Z 


I06  KINETICS    OF   A   PARTICLE.  [195. 

These  equations  (4),  in  connection  with  the  two  conditions  (i), 
are  sufficient  to  determine  the  five  quantities  x,  y,  z,  X,  /JL  as 
functions  of  the  time;  the  values  of  x,  y,  z  so  found  give  the 
position  of  the  particle  at  any  time,  while  X,  p  can  be  shown 
to  determine  the  pressure  on  the  curve. 

195.  To  find  the  reaction  A^  of  the  curve,  let  us  compare  the 
equations  (4)  with  the  equations  of  Art.  161.  It  appears  at 
once  that  the  forces  X\  V,  Z'  that  would  replace  the  condi- 
tions (i),  i.e.  the  components  of  the  reaction  A7"  of  the  constrain- 
ing curve,  are 


whence 

.  (5)- 


These  equations  determine  the  magnitude  and  direction  of  the 
reaction  N,  as  soon  as  X  and  /JL  are  found. 

196.  Let  us  now  combine  the  equations  (4)  according  to  the 
principle  of  kinetic  energy  ;  that  is,  multi  ply  them  by  dx,  dy, 
dz,  and  add.  The  left-hand  member  becomes,  of  course,  the 
exact  differential  d(^mv^).  The  right-hand  member, 


will  in  general  contain  terms  depending  on  the  reaction  of  the 
surface;  in  other  words,  in  the  actrial  displacement  ds  =  (d^  + 
.dy^+dz*)^  of  the  particle  the  reaction  of  the  moving  curve  will 
in  general  do  work. 

197.  Only  in  the  particular  case  when  the  curve  is  fixed  will  1 
the  work  of  the  reaction  be  zero;  for  in  this  case  the  condi- 
tional equations  (i)  do  not  contain  the  time  explicitly,  and  their  1 
complete  differentiation  gives  the  relations 

z  =  o,     rxdx  +      dy  +  ^rzdz  =  O, 


I99-]  VARIABLE   CONSTRAINTS.  107 

which  show  that  the  coefficients  of  \  and  /JL  in  the  equation 
of  kinetic  energy  vanish. 

Hence,  for  motion  on  a  fixed  curve  we  have 

d(\  mv2)  =  Xdx  +  Ydy  +  Zdz,  (6) 

which  agrees  with  the  equation  (7)  of  Art.  173,  considering 
that  the  frictional  resistance  is  supposed  to  be  included  among 
the  forces  X,  Y,  Z. 

198.    In  the  general  case,  the  complete  differentiation  of  the 
equations  (i)  gives 


and  the  equation  of  kinetic  energy  -for  motion  on  a  moving  or 
variable  curve  becomes 


(7) 

The  distinction  between  the  virtual  displacement  Ss  along 
the  curve  in  its  position  at  the  time  t  and  the  actual  displace- 
ment ds  of  the  particle  along  the  moving  curve  should  be 
clearly  understood.  The  virtual  displacement  &s  =  PP'  joins 
the  position  P  (x,  y,  z)  of  the  particle  at  the  time  /  to  a  point 
P'  (x+§x,  y  +  ty,  z  +  §z),  which  is  on  the  curve,  and  infinitely 
near  to  P  at  the  time  /,  while  the  actual  displacement  ds  =  PP" 
joins  P  (x,  y,  z)  to  the  position  P"  (x+dx,  y  +  dy,  z-\-dz)  of  the 
particle  at  the  time  t  +  dt\  P"  lies,  therefore,  on  the  position 
that  the  curve  has,  not  at  the  time  t,  but  at  the  time  t  +  dt. 
The  reaction  N  of  the  curve  at  the  time  t  is  normal  to  Ss, 
but  not  to  ds. 

199.   One  Constraint.     Let 

$(x,y,  z,  *)=o  (8) 

be  the  equation  of  the  surface  on  which  the  particle  is  assumed 
to  remain  throughout  its  motion.  The  reaction  N  of  this  sur- 
face will  do  no  work  if  the  displacement  Ss  be  taken  along  the 


108  KINETICS   OF  A   PARTICLE.  [200. 

position  of  the  surface  at  a  given  time  t.  In  other  words,  to 
obtain  a  displacement  Ss  compatible  with  the  condition  (8), 
its  components  &r,  By,  Sz  should  satisfy  the  condition 

<k&T+<k^  +  <k&?  =  o,  (9) 

obtained  by  differentiating  the  equation  (8)  with  respect  to 
the  co-ordinates. 

200.  By  means  of  the  relation  (9),  one  of  the  displacements 
&r,  By,  Sz  can  be  eliminated  from  the  general  equation  of 
motion  (2);  the  two  remaining  displacements  will  then  be 
independent,  and  their  coefficients  can  therefore  be  equated 
to  zero  separately. 

The  elimination  is  again  conveniently  effected  by  the  method 
of  indeterminate  multipliers.  Multiplying  equation  (9)  by  an 
indeterminate  factor  X,  and  adding  it  to  equation  (2),  we  find 
the  single  equation 


in  which  the  arbitrary  quantity  X  can  be  so  selected  as  to  make 
the  coefficient  of  one  of  the  three  displacements  vanish.  The 
other  two  displacements  being  arbitrary,  their  coefficients  must 
also  vanish.  The  last  equation  can  therefore  be  replaced  by 
the  following  three  : 


my  =  Y+\$y,  mz  =  Z+\<}>t,  (10) 

which,  in  connection  with  the  given  condition  (8),  fully  deter- 
mine the  problem  ;  for  they  are  sufficient  for  finding  x,  y,  z,  and 
X  as  functions  of  t. 

201.   Just  as  in  Art.  195,  it  follows  that  the  components  of 
the  reaction  N  of  the  surface  are 


whence  7Vr  =  xV<#>x2  +  ^2  +  <#>z2.  (n) 


204-]  VARIABLE   CONSTRAINTS. 

202.    If   the  equations   (10)   be   combined   according   to   the 
principle  of  kinetic  energy,  we  find 


where  again  the  coefficient  of  X  vanishes  only  when  the  sur 
face  is  fixed,  in  which  case 


(12) 

while  in  the  general  case  of  a  moving  or  variable  surface  we 
have 

(13) 


203.  Plane  Motion.  If  a  particle  be  constrained  to  move  in  a 
plane  curve  under  the  action  of  forces  lying  in  the  plane  of  the 
curve,  d'Alembert's  principle  gives  the  equation  of  motion 

and  the  equation  of  the  curve 

<f>(x,  y,  /)=o  (l$) 

gives  by  differentiation  for   a  virtual  displacement  Ss  on  the 
curve  at  a  given  time  /, 

Hence,  proceeding  as  in  Art.  200,  the  equations  of  motion  can 
be  written  in  the  form 

mx = X  +  \<£>x,     my-=  Y-\-\<f)v,  (17) 

while  the  normal  reaction  of  the  curve  is 

(18) 


204.  The  process  of  solution  is  now  as  follows  for  the  case  of 
plane  motion.  Differentiate  the  equation  of  condition  (15), 
which  holds  at  any  time,  with  respect  to  the  time,  remembering 
that  x  and  y  are  functions  of  the  time  ;  this  gives  : 

o.  (19) 


IIO  KINETICS   OF   A   PARTICLE.  [205. 

Differentiating  again,  we  find 


+  2X<l>tx  +  2>(£,y  +  (/>„  =  O.  (20) 


If  in  this  last  equation  the  values  of  Jr,  y  be  substituted  from 
(17),  we  have  a  linear  equation  for  X.  The  value  of  X  thus 
obtained  can  then  be  introduced  into  the  equations  of  motion 
(17)  ;  and  it  only  remains  to  integrate  these  equations. 

This  integration  will  often  be  facilitated  by  introducing  new 
variables  for  x,  y. 

205.  A  particle  moves  without  friction  in  a  straight  tube  which 
revolves  uniformly  in  a  horizontal  plane  about  one  of  its  points.  De- 
termine its  motion. 

To  illustrate  the  application  of  the  general  methods,  we  shall  solve 
this  problem  completely,  first  without  the  use  of  indeterminate  multi- 
pliers, and  then  with  their  aid,  although  the  problem  is  so  simple  that  it 
might  be  solved  without  applying  these  general  methods,  as  will  be 
pointed  out  below. 

As  the  weight  of  the  particle  is  balanced  by  the  vertical  reaction  of 
the  tube,  we  have  a  case  of  plane  motion  with  X  =  o,  Y—  o.  Hence 
d'Alembert's  equation  (14)  becomes 

x8x  +  y8y=o.  (21) 

If  we  take  as  origin  the  point  O  about  which  the  tube  rotates,  the  con- 
straining curve  is  a  straight  line  through  the  origin  y  =  x  tan  0,  where 
6  =  to/,  to  being  the  constant  angular  velocity  of  the  tube  and  the  axis  of 
x  coinciding  with  the  initial  position  of  the  tube  at  the  time  /=  o.  Hence 

x  =  r  cos  to/,     y  =  r  sin  to/  ;         ^ 

8x  =  Br  cos  to/,     By  =  Br  sin  to/  ; 

(22) 
x  =  r  cos  to/  —  tor  sin  to/,     y=  r  sin  to/  +  tor  cos  to/; 

x=r  cos  <o/—  2  tor  sin  to/—  toV  cos  to/,     y=  r  sin  to/+  2  tor  cos  to/—  to2r  sin  to/. 

Substituting  these  values  of  x,  y  and  8x,  By  into  the  equation  of  motion 
(21),  we  find  after  reduction 

r—  wV=o.  (23) 

As  mentioned  above,  this  equation  might  have  been  derived  directly 
by  considering  that  the  acceleration  along  the  tube  is  due  to  the  cen- 
trifugal force  alone  (see  Art.  1  70)  . 


2o;.]  VARIABLE   CONSTRAINTS.  Iir 

206.   The  general  integral  of  equation  (23)  is 


r  = 

If  r=  r0  and  r  —  vQ  when  /  =  o,  we  have  r0  =  ^  +  <r2,  #0  =  <»(^i  —  <r2)  ; 
hence 

2  or  =  (<or0  +  Vo)e<*+(u>ro  —  z>0  )*""*.  (24) 

With  #0  =  0,  r0=i,  this  equation  represents  a  common  catenary.. 
If  VQ  =  r0o>,  the   equation   reduces  to  the  form  r  =  rQeMt,  whence  /  = 


The  minimum  of  r  in  (24)  occurs  for 


2o)        cor0  -f-  VQ 

its  value  is  r^  =  VV02  —  (#0/co)2.     It  is  easy  to  see  that  such  a  minimum 
can  occur  only  when  z>0  is  negative  and  >  wr0  numerically. 

207.   To  apply  the  method  of  indeterminate  multipliers  to  our  prob- 
lem, let  the  equation  of  the  tube  be  written  in  the  form 


$(x,  y,  /)  =  .#  cos  to/  —  j^sina)/t=  o.  (25) 

Then  we  have  <f>x  =  cos  o>/,  <j>,  =  —  sin  <o/,  </>,=  —  v(x  sin  <o/4-jy  cos  <o/)  ; 
hence  equation  (16)  assumes  the  form 

Bx  cos  w/  —  By  sin  to/  =  o  ; 
and  the  equations  of  motion  (17)  are 

mx  =  A  cos  CD/,    #2)}  =  —  A  sin  to/.  (26) 

We   have   also   <f>xx  =  o,   ^  =  <$>yx  =  o,  <^>yy  =  o,   ^x  =  ^  =  —  w  sin  w/r 
^  =  ^yt  =  —  w  cos  w^>    <#>«  =  —  °>2(*  cos  w/—  j^  sin  co/)  =  o.     Hence,  by 
' 


—  J-5  3r  cos  to/—  j  sin  co/—  2  our  sin  to/—  2  toj;  cos  to/  =  o. 
dr 

Substituting  in  this  equation  the  values  of  x,y  from  (26),  we  find  the 
linear  equation  for  X  which  gives 

—  =  2  to  (x  sin  to/  -f^  cos  co/)  sec  2  to/. 
m 

Introducing  this  value  into  the  equations  (26),  we  have  the  differen- 
tial equations  of  our  problem  in  the  form 

x  =  2(a(x  sin  co/  +  y  cos  to/)  —  —  ^—  ,  y=  —  2to(^sinco/-f-j;cosco/) 


COS  2tO/     "  COS  2CO/ 


112  KINETICS   OF   A   PARTICLE.  [208. 

Their  integration  can  best  be  performed  by  introducing  the  radius 
vector  r  by  means  of  the  relations  (22).  Multiplying  the  equations 
respectively  by  cosw/  and  sinco/,  and  adding,  we  find  that  the  right- 
hand  member  vanishes,  and  we  have 

x  cos  w/  +  y  sin  w/  ==  o, 
or,  substituting  for  x  and  y  their  values  from  (22), 

r  —  w?r  =  o, 
which  agrees  with  (23),  Art.  205. 

208.   For  the  pressure  on  the  curve,  we  have,  by  ( 18),  since  <f>s?-t-<j>f=  i, 
N=  \  =  _^!L  (x  sin <o/  + j>  cos  o>/). 

COS  2  CO/ 

Substituting  from  (22)  and  (24),  and  reducing,  we  find 

N=  ///a>[  (o»o  +  z>o)^(i  +  tan  2  o>/)  +  (o>r0  —  VQ)  <?-"*(  i  —  tan  2  co/)]. 

209.   Exercises.* 

(1)  A  particle  subject  to  gravity  moves  without  friction  in  a  straight 
tube  which  revolves  uniformly  in  a  vertical  circle.      Find  the  distance  r 
of  the  particle  from  the  centre  of  rotation  at  any  time  /. 

(2)  A  particle  moves  without  friction  in  a  circular  tube  which  rotates 
uniformly  in  a  horizontal  plane  about  a  point  O  in  its  circumference. 
If  the  particle  is  at  the  time  /=o  at  rest  at  the  end  of  the  diameter 
passing  through  O,  what  is  its  position  at  any  time  /  ? 

(3)  A  particle  moves   in  a  horizontal  circular  tube  whose  radius 
increases  proportionally  to  the  time.      At  the  time  /  =  o  the  radius  is  a, 
and  the  particle  has  a  velocity  VQ  perpendicular  to  the  radius.     Find  the 
position  and  velocity  at  any  time  /. 


*  These  examples  are  taken  from  Walton's  Collection  (referred  to  in  Art.  159), 
pp.  401-406. 


2ii.]  LAGRANGE'S   EQUATIONS.  H3 

V.    Lagrange  s  Form  of  the  Equations  of  Motion. 

210.  It  has  been  shown  in  Arts.  180,  181,  how  the  equations 
of  motion  on  a  fixed  curve  can  be  made  to  depend  on  a  single 
variable  q,  and  in  Art.  191   how  the  motion  on  a  fixed  surface 
can  be  expressed  by  means  of  two  variables  qv  q^.     By  apply- 
ing this  idea,  and  by  introducing  the  kinetic  energy  T  and  its 
derivatives,  the  equations  of  motion  of  a  particle  with  or  without 
conditions  can  be  put  into  a  remarkably  compact  form,  which 
was  first  devised   by  Lagrange  for   the   general   equations    of 
motion  of  a  system  of  n  particles  (comp.  Arts.  385-394).     We 
proceed  to  establish  these  equations,  first  for  the  case  of  motion 
on  a  variable  curve,  then  for  motion  on  a  variable  surface,  and 
finally  for  a  free  particle. 

211.  Particle   Subject  to  Two   Conditions.      As  shown  in  Art. 
194  (comp.  Art.    192),   the  equations  of  motion  in   Cartesian 
co-ordinates  can  be  written  in  the  form 


mX  =  X+  \<f)x 

my  =  F+x<k  +  ^,,  (i) 

mz  =  Z  4-  \<f)z  +  A^,» 
if  the  equations  of  condition  are 

<£(*,  y,  2,  /)=o,  -f(>:,  y,  z,  t}=Q.  (2) 

The  single  variable  q,  that  determines  the  position  of  the 
particle  on  the  curve,  is  called  the  Lagrangian,  or  generalized, 
co-ordinate  of  the  particle.  The  Cartesian  co-ordinates,  x,  y,  z, 
are  functions  of  the  Lagrangian  co-ordinate  q,  and  of  the  time 
/,  say 

*=/i('.  <?)>  y=A(*>  ?)>  *=/*(*>  ?)•  (3) 

To  introduce  q  in  the  place  of  x,  y,  z,  we  shall  need  the 
derivatives  x>  j/,  z.  The  first  of  the  equations  (3)  gives 


PART   III  —  8 


KINETICS    OF   A    PARTICLE.  [212. 

so  that  x  can  be  regarded  as  a  function  of  t,  q,  and  q  =  dq/dt. 
Hence  we  have 

dq     dqdt      dq2   '      dq      dq 
In  the  former  of  these  expressions,  the  right-hand  member  can 

be  put  into  the  more  compact  form =£i,  as  is  easily  verified 

dt  dq 

by  carrying  out  the  indicated  differentiation  with  respect  to  t. 
As  similar  results  hold  for  y  and  z,  we  have 


/\ 
dq~dtdq      dq~dtdq 


__ 

'    ~~ 


' 


dq      dq  dq      dq  dq      dq 

212.  Let  us  now  add  the  equations  of  motion  (i)  after  multi- 
plying them  by  dfjdq,  df^/dq,  dfjdq.  The  coefficient  of  X  in 
the  resulting  equation,  viz. 


^_,^,^ 

dx   dq      dy  dq      dz    dq' 


is  equal  to  zero,  since  it  is  evidently  proportional  to  the  cosine 
of  the  angle  made  at  a  given  time  by  the  tangent  to  the  curve 
(3)  with  the  normal  to  the  surface  </>=o.  For  a  similar  reason, 
the  coefficient  of  /j,  vanishes  ;  and  the  resulting  equation  is 

m(xdA+ydA+zSA\=Q,  (6) 

V   dq          dq         dqj 
if,  as  in  Arts.  180,  181,  we  put  for  shortness 


. 

dq          dq          dq 

This  quantity  Q  can  evidently  be  expressed  as  a  function  of 
q,  and  q. 

The  equation  (6)  can  also  be  written  in  the  form 


m  — 


2i3.]  LAGRANGE'S   EQUATIONS.  U5 

as  appears  by  carrying  out  the  indicated  differentiations  with 
respect  to  /,  and  if  we  now  make  use  of  the  relations  (4)  and  (5), 
our  equation  assumes  the  form 

d     .  te     .  d        .  ds\  .  dx        Q        .  dz 


The  quantities  in  the  two  parentheses,  each  multiplied  by  m, 
are  easily  recognized  as  the  partial  derivatives  with  respect  to 
q  and  q  of  the  kinetic  energy 


hence  the  equation  reduces  to  the  form 

±*L-*L=Q  (7) 

dt  dq       dq      V> 

known  as  the  (second)  Lagrangian  form  of   the   equation   of 
motion  of  a  particle  constrained  to  a  curve. 

213.   Particle  Subject  to  One  Condition.     By  Art.  200,  the  equa- 
tions of  motion  are 


(8) 

with  the  condition 

$(pty,z,t)=o.  (9) 

Let  the  two  generalized  co-ordinates  q^  q^  be  connected  with 
the  Cartesian  co-ordinates  by  the  equations 


The  first  of  these  equations  gives 


r 

dt      dq± 

hence,    regarding  x    as    a    function   of    /,  q^    q^, 
)  we  find  : 


KINETICS    OF   A   PARTICLE. 


[214. 


The  right-hand  member  of   the  former  of  these  equations  is 

equivalent  to  -j  ~     As  similar  relations  hold  for  y  and  z,  we 
dtdql 

find  again  the  relations  (4)  and  (5),  with  ql  substituted  for  q. 
It  can  be  shown  in  the  same  way  that  these  relations  also  hold 
if  <72  be  written  for  q. 


214.    Let  us  now  multiply  the  equations  (8)  by  5/i/d^, 
!,  and  add  them.     This  gives 


where 


Similarly,   multiplying    (8)    by    dfi/dg2, 
adding,  we  find 


2'    5/3/%2>    and 


m 


where 


Each  of  these  equations  (n)  and  (12)  can  be  treated  by  the 
method  used  in  Art.  212,  and  we  find  as  the  final  equations  of 
motion  on  the  surface  (9)  in  the  Lagrangian  form : 


dT 


d_ 

dt 


(13) 


215.  Free  Particle.  In  this  case  three  variables  qv  q^  <?3  are 
required  to  determine  the  position  of  the  particle.  If  the 
expressions  of  x,  yy  z  in  terms  of  these  new  variables  do  not 
contain  the  time  explicitly,  the  introduction  of  the  new  varia- 
bles consists  merely  in  a  change  of  co-ordinates.  If  they  do 
contain  the  time,  i.e.  if  we  have 


the  new  system  of  co-ordinates  is  a  moving  system. 


2 1 6.]  LAGRANGE'S    EQUATIONS.  U^ 

The  relations  (4),  (5)  can  again  be  shown  to  hold  for  each 
of  the  three  Lagrangian  co-ordinates  gv  q^  q%. 

If  the  equations  of  motion 

mx=X,    my=Y>    m'z  —  Z  (15) 

be  multiplied  by  3/i/d^,  d/2/d<?i,  dfjdq^  and  added,  we  find 


where  '    Q^X^+Y^Z^ 

dql         dgl         dql 

By  the  method  of  Art.  212,  equation  (16)  reduces  to  the  form 

^ar_ar_ 

Similarly  we  find 

Cv     (j  -L  U  J.  S-* 

77/  ^       ^T==^2» 


The  three  equations  (17)  and  (18)  are  the  Lagrangian  equations 
of  motion  of  a  free  particle. 

216.    If  there  exists  a  force-function   U  for  the  forces  X,  F, 

Z,  i.e.  if 

Y    dU      v    BU      -    dU 
JL=—-t      y=——,     <£  =  —- 

dx  dy  dz 

we  have 


_          +         ^  +          ,= 
1      dx  d^1      dy  dql      dz  d^l 


and  similarly 

3U 


In  this  case,  one  of   the  three  equations   (17),  (18)  can  be 
replaced  by  the  equation  of  kinetic  energy 


where  h  is  a  constant. 


Il8  KINETICS    OF   A   PARTICLE.  [217. 

217.  The  general  theory  of  the  constrained  motion  of  a  particle  is 
treated  with  special  care  in  the  works  of  Schell,  Budde,  and  Appell, 
referred  to  in  Art.  159.  In  Appell's  first  volume,  pp.  445-517,  the 
student  will  find  instructive  examples  of  the  application  of  Lagrange's 
equations.  For  more  elementary  problems,  as  also  for  the  interesting 
theories  of  brachistochrones  and  tautochrones,  the  reader  is  referred, 
besides  the  works  just  named,  to  the  text-books  of  Tait  and  Steele, 
Besant,  Price,  and  Walton's  Problems  (see  Art.  159). 


2 1 9-]  GENERAL   PRINCIPLES.  1 19 


CHAPTER   VI. 

KINETICS    OF    A    RIGID    BODY. 

I.    General  Principles. 

*  218.  In  kinetics  the  term  rigid  body  means  any  system  or 
aggregate  of  mass-particles  whose  mutual  distances  remain 
invariable.  A  rigid  body  may  therefore  consist  of  a  finite 
number  of  rigidly  connected  particles  or  of  a  continuous  mass 
of  one,  two,  or  three  dimensions.  Its  motion  depends  not  only 
on  the  forces  acting  on  the  body,  but  also  on  the  way  in  which 
the  mass  is  distributed  throughout  the  body.  * 

In  the  present  section  the  rigid  body  is  assumed  to  be  free 
unless  the  contrary  be  stated  explicitly. 

219.  Let  us  consider  any  one  particle  m  of  the  body ;  at  any 
time  /,  let  j  be  its  acceleration  and  F  the  resultant  of  all  the 
forces  acting  on  the  particle.  Then  the  motion  of  this  particle 
(see  Arts.  35,  67)  is  determined  by  the  equation 

mj—F.  (i) 

It  should  be  noticed  that  among  the  forces  acting  on  the 
particle  are  included  not  only  those  external  forces  acting  on 
the  rigid  body  that  happen  to  be  applied  at  m,  but  also  the 
.so-called  internal  forces  which  would  replace  the  rigid  con- 
nection of  the  particle  m  with  the  rest  of  the  body. 

If,  at  the  time  t,  x,  y,  2  are  the  co-ordinates  of  the  particle 
•m  with  respect  to  a  fixed  set  of  rectangular  axes,  then  the 
components  of  its  velocity  v  may  be  denoted  by  x,  y,  z\  those 
•of  its  acceleration  j  by  xt  y,  z*  And  if  the  components  of  F 

*  Here  again  we  shall  use  this  so-called  fluxional  notation,  according  to  which 
derivatives  with  respect  to  the  time  are  denoted  by  dots;  see  the  foot-note  to  Art.  183. 


120  KINETICS   OF   A   RIGID   BODY.  [220. 

along  the  same  axes  are  X,    Y,  Z,   the  equation   (i)   can  be 
replaced  by  the  following  three  : 


=o,    —my  +  Y=o,    —  mz  +  Z=o.  (2) 

Such  a  set  of  three  equations  can  be  written  down  for  each 
particle  ;  hence,  if  the  body  consist  of  n  particles,  there  would 
be  in  all  3^  equations. 

220.  For  the  solution  of  particular  problems  these  3/2  equa- 
tions are  of  little  use,  not  only  because  their  number  would 
in  general  be  very  great  and  may  even  be  infinite,  but  mainly 
because  the  forces  X,    Y,  Z  include  the   unknown    reactions 
between  the  particles.     It  is,  however,  possible  to  deduce  cer- 
tain general  propositions  from  these  equations. 

The  3«  equations  express  the  equilibrium  of  the  system 
formed  by  all  the  forces,  both  internal  and  external,  acting  on 
the  particles,  and  the  reversed  effective  forces.  To  apply  the 
principle  of  virtual  work  to  this  system,  let  us  multiply  the 
three  equations  (2)  by  the  components  Sx,  Sj/,  §z  of  some  virtual 
displacement  of  the  particle  m  ;  let  the  same  thing  be  done 
for  every  other  particle  of  the  body  ;  and  let  all  the  resulting 
equations  be  added  : 

2(-0*jr+-Y)&r+2(-»*y+  F)fy  +  2(-*«2+Z)&sr=o.      (3) 

221.  It  is  important  to  notice   that   the   internal  reactions 
between   the   particles   which   make   the   body  rigid  occur  in 
pairs    of    equal    and   opposite   forces,    and   form,   therefore,   a 
system  which  is  in  equilibrium  by  itself.     Hence,  while  these 
internal  forces  enter  into  the  equations  (2),  they  do  not  appear 
in  equation  (3),  since  the  equal  and  opposite  forces  cancel  in 
the  summation.     Thus,  equation  (3)  expresses  that  the  external 


Derivatives  were  called  fluxes  by  Newton;  thus  the  component  of  the  acceleration 
of  a  point  in  any  direction  is  the  time-flux  of  its  velocity  in  that  direction;  the  com- 
ponent of  its  effective  force  in  any  direction  is  the  time-flux  of  its  momentum; 
and  so  on.  « 


223.]  GENERAL   PRINCIPLES.  I2i 

forces  acting  on  the  rigid  body  and  the  reversed  effective  forces 
form  a  system  in  equilibrium  ;  and  this  is  d'Alembert's  Principle 
for  the  rigid  body. 

It  must,  however,  not  be  forgotten  that  the  displacements  $>x, 
Sj>,  &z  should  be  so  selected  as  to  be  compatible  with  the  nature 
of  the  rigid  body;  i.e.  with  the  conditions  that  the  distances 
between  trie  particles  should  not  be  disturbed. 

222.  The  number  of  conditions  expressing  the  invariability 
of  the  distances  between  n  particles  is    3/2  —  6.     For  if  there 
were   but   3   particles,  the  number  of   independent    conditions 
would  evidently  be  3  ;  for  every  additional  particle,  3  additional 
conditions  are  required.     Hence,  the  total    number   of   condi- 
tions is  3  +  3(«-3)  =  3^-6. 

It  follows  that  if  a  rigid  body  be  subject  to  no  other  con- 
straining conditions,  the  number  of  its  equations  of  motion 
must  be  $n  —  ($n—  6)  =6.  Hence,  a  free  rigid  body  has  six 
independent  equations  of  motion.  (Comp.  Part  I.,  Art.  37.) 

223.  The  six  equations  of  motion  of  the  rigid  body  can  be 
obtained  as  follows. 

Imagine  the  equations  (2),  viz. 

mx=X,    my=  Y,    mz  =  Z, 

written   down   for   every   particle,  and   add  the  corresponding 

equations.     This  gives  the  first  3  of  the  6  equations  of  motion: 

^mx=^X1    ^my  =  ^Yy    2mz  =  2Z.  (4) 

As  the  internal  forces  cancel  in  the  summation,  the  right-hand 
members  of  these  equations  represent  the  components  Rx,  Ry,  Rg 
of  the  resultant  R  of  all  the  external  forces  acting  on  the  body. 
The  left-hand  members  can  be  put  into  the  form  d$mx)/dt, 
d@.my)/dt,  d(*Lmz)/dt  ;  these  are  the  time  derivatives  or  fluxes 
of  the  sums  of  the  linear  momenta  of  all  the  particles  parallel 
to  the  axes.  The  equations  (4)  can  therefore  be  written  in  the 
form 

Rr  (5) 


a  ,, 

dt  dt  dt 


I22  KINETICS   OF   A   RIGID   BODY.  [224. 

The  axes  of  co-ordinates  are  arbitrary.  Hence,  if  we  agree  to 
call  linear  momentum  of  the  body  in  any  direction  the  algebraic 
sum  of  the  linear  momenta  of  all  the  particles  in  that  direc- 
tion, the  equations  (5)  express  the  proposition  that  the  rate'  at 
which  the  linear  momentum  of  a  rigid  body  in  any  direction 
changes  with  the  time  is  equal  to  the  sum  of  the  components  of 
all  the  external  forces  in  that  direction. 

224.  Let  us  now  combine  the  second  and  third  of  the  equa- 
tions (2)  by  multiplying  the  former  by  2,  the  latter  by  y,  and 
subtracting  the  former  from  the  latter.  If  this  be  done  for 
-each  particle,  and  the  resulting  equations  be  added,  we  find 
^m(y'z  —  zy)  =  ^(yZ—  zY).  Similarly,  we  can  proceed  with  the 
third  and  first,  and  with  the  first  and  second  of  the  equations 
<2).  The  result  is  : 


(6) 

Here  again  the  internal  forces  disappear  in  the  summation, 
so  that  the  right-hand  members  are  the  components  Hx,  Hy,  H, 
of  the  vector  H  of  the  resultant  couple,  found  by  reducing 
all  the  external  forces  for  the  origin  of  co-ordinates.  The 
left-hand  members  are  the  components  of  the  resultant  couple 
of  the  effective  forces  for  the  same  origin. 

We  can  also  say  that  the  right-hand  members  are  the  sums 
of  the  moments  of  the  external  forces  about  the  co-ordinate 
axes  (Part  II. ,  Art.  213),  while  the  left-hand  members  repre- 
sent the  moments  of  the  effective  forces  about  the  same  axes. 
The  latter  quantities  are  exact  derivatives,  as  shown  in  Arts. 
87  and  91.  The  equations  (6)  can  therefore  be  written  in 
the  form 


As  explained  in  Arts.  89  and  92,  the  quantity  m(yz—zy)  is 
called  the  angular  momentum  (or  the  moment  of  momentum] 


225.]  GENERAL   PRINCIPLES.  I23 

of  the  particle  m  about  the  axis  of  x.  We  may  now  agree  to 
call  the  quantity  ^m(yz-zy)  the  ang^ilar  momentum  of  the 
body  about  the  axis  of  x>  just  as  ^mx  is  the  linear  momentum 
of  the  body  along  this  axis ;  and  similarly  for  the  other  axes. 
The  meaning  of  the  equations  (7)  can  then  be  stated  as  follows  : 
The  rate  at  which  the  angular  momentum  of  a  rigid  body  about 
any  axis  changes  with  the  time  is  equal  to  the  sum  of  the 
moments  of  all  the  external  forces  about  this  line. 

The  equations  (4)  and  (6),  or  (5)  and  (7),  are  the  six  equa- 
tions of  motion  of  the  rigid  body.  The  three  equations  (4)  or 
(5)  may  be  called  the  equations  of  linear  momentum,  while  (6) 
or  (7)  are  the  equations  of  angular  momentum. 

225.  The  equations  (4)  and  (6)  can  also  be  derived  from  the  equa- 
tion (3),  which  expresses  d'Alembert's  principle,  by  selecting  for  £#, 
By,  Bz  convenient  displacements. 

Thus,  the  rigidity  of  the  body  will  evidently  not  be  disturbed  if  we 
give  to  all  its  points  equal  and  parallel  infinitesimal  displacements,  since 
this  merely  amounts  to  subjecting  the  whole  body  to  an  infinitesimal 
translation.  Equation  (3)  can  in  this  case  be  written 

&#:§(-  mx  +  X)  +  8y2(  —  my+  Y)  +  Bz%(  —  mz  +Z)  =  o, 

and  is  therefore  equivalent  to  the  three  equations  (4),  since  8.*,  By,  Bz 
are  independent  and  arbitrary. 

Again,  let  the  body  be  subjected  to  an  infinitesimal  rotation  of  angle 
BO  about  any  line  /. 

As  shown  in  Art.  293  of  Part  I.,  the  linear  velocities  of  any  point 
(x,  y,  z)  of  a  rigid  body,  due  to  a  rotation  of  angular  velocity  o>  =  80/8/ 
about  any  line  /  are,  if  eo,.,  o>y,  wz  denote  the  components  of  o> : 

x  =  o)yZ  —  <azy,  y  =  (DgX  —  <DXZ,   z  =  <Dxy—  oyr. 

Hence,  putting  eox8/=80x,  o>yS/=80y,  <u,8/=80,,  we  have  for  the 
displacements  of  the  point  (x,  y,  z) ,  due  to  a  rotation  of  angle  80, 

Bx  =  zBOy-yBOz,   By  =  xBez-zB9x,   Bz  =  y S0Z- x Wr 

If  these  values  be  introduced  into  d'Alembert's  equation  (3)  and  the 
terms  in  S0Z,  B0y,  Wz  be  collected,  it  assumes  the  form 

S0X2[-  m(yz  -  zy)  +yZ-zY]  +  S0y2[-  m(zx  -  xz)  +  zX-  xZ~\ 
+  80,5  [—  m(xy  —yx)  +  xY—yX']=o ; 


I24  KINETICS   OF   A   RIGID   BODY.  [226. 

as  80X,  SOy,  80Z  are  independent  and  arbitrary,  their  coefficients  must 
vanish  separately,  and  this  gives  the  equations  (6). 

226.  The  equations  of  linear  momentum,  (4)  or  (5),  admit  of 
a  further  simplification,  owing  to  the  fundamental  property  of 
the  centroid.  By  Part  II.,  Art.  13,  the  co-ordinates  x,  j>,  z  of  the 
centroid  satisfy  the  relations 

MX  —  ^mxy    My  =  ^my,    Mz = 2mz, 

where  M=  2m  is  the  whole  mass  of  the  body.     Differentiating 
these  equations,  we  find 

MX = 2, mx,   My  =  ^myy    Mz  =  2 
and  MX  =  ^mx,    My  = 


where  x,  y,  z  are  the  components  of  the  velocity  v,  and  x,  y, 
those  of  the  acceleration/,  of  the  centroid. 

The  equations  (4)  or  (5)  can  therefore  be  reduced  to  the  form 

Mx=  —Mx—Rx,     My  =  —  My  =  Ry,     Mz  =  ~rM~z  —  Rg)     (8) 

whence  Mj=  -  Mv =R\  (9) 

at 

i.e.  if  the  whole  mass  of  the  body  be  regarded  as  concentrated 
at  the  centroid,  the  effective  force  of  the  centroid,  or  the  time- 
rate  of  change  of  its  momentum,  is  equal  to  the  resultant  of  all 
the  external  forces.  It  follows  that  the  centroid  of  a  rigid  body 
'moves  as  if  it  contained  the  whole  mass,  and  all  the  external  forces 
were  applied  at  this  point  parallel  to  their  original  directions. 

227.  If,  in  particular,  the  resultant  R  vanish  (while  there  may 
be  a  couple  H  acting  on  the  body),  we  have  by  (8)  and  (9) 
y=o;  hence  v  =  const. ;  i.e.  if  the  resultant  force  be  zero,  the 
centroid  moves  uniformly  in  a  straight  line. 

This  proposition,  which  can  also  be  expressed  by  saying  that, 
if  R  =  o,  the  momentum  Mv  of  the  centroid  remains  constant, 
or,  using  the  form  (5)  of  the  equations  of  motion,  that  the  linear 
momentum  of  the  body  in  any  direction  is  constant,  is  known 


]  GENERAL   PRINCIPLES.  125 

~he  principle  of  the  conservation  of  linear  momentum,  or  the 

principle  of  the  conservation  of  the  motion  of  the  centroid. 

228.  Let  us  next  consider  the  equations  of  angular  momen- 
tum, (6)  or  (7).  To  introduce  the  properties  of  the  centroid, 
let  us  put  x—  x=%,  y—y  =  fn,  z  —  z  =  £,  so  that  £,  77,  f  are  the 
co-ordinates  of  the  point  (x,  y,  z)  with  respect  to  parallel  axes 
through  the  centroid.  The  substitution  of  x=x+%,  jj/  =3/4-77, 
their  derivatives  into  the  expression  yz—zy  gives 


To  form  ^m(yz-zy)  we  must  multiply  by  m  and  sum  through- 
out the  body  ;  in  this  summation,  y,  ~z,  y,  ~z  are  constant  and, 
by  the  property  of  the  centroid, 
2m=o.  Hence  we  find 


The  second  term  in  the  right-hand  member  is  the  angular 
momentum  of  the  centroid  about  the  axis  of  x  (the  whole  mass 
M  of  the  body  being  regarded  as  concentrated  at  this  point), 
while  the  first  term  is  the  angular  momentum  of  the  body  about 
a  parallel  to  the  axis  of  x,  drawn  through  the  centroid. 

Similar  relations  hold  for  the  angular  momenta  about  the 
axes  of  y  and  z  ;  and  as  these  axes  are  arbitrary,  we  conclude 
that  the  angular  momentum  of  a  rigid  body  about  any  line  is 
equal  to  its  angular  momentum  about  a  parallel  through  the 
centroid  plus  the  angular  momentum  of  the  centroid  about  the 
former  line. 

229.    Differentiating  the  above  expression,  we  find 


The  first  of  the  equations  (7)  can  therefore  be  written 

| 
dt 


I26  KINETICS    OF   A   RIGID    BODY.  [230. 

Now,  if  at  any  time  t  the  centroid  were  taken  as  origin,  so  that 
3/  =  o,  i=o,  this  equation  would  reduce  to  the  form 

£?,m(rt-tt)=jrf 

which  is  entirely  independent  of  the  co-ordinates  of  the  cen- 
troid. On  the  other  hand,  wherever  the  origin  is  taken,  if 
the  centroid  were  a  fixed  point,  the  same  equation  would 
be  obtained. 

Similar  considerations  apply  of  course  to  the  other  two 
equations  (7).  It  follows  that  the  motion  of  a  rigid  body  relative- 
to  the  centroid  is  the  same  as  if  the  centroid  were  fixed. 

'  230.  If,  in  particular,  the  resultant  couple  H  be  zero  for  any 
particular  origin  O  (which  will  be  the  case  not  only  when  all 
external  forces  are  zero,  but  also  when  the  directions  of  all 
forces  pass  through  the  point  O),  the  equations  (7)  can  be 
integrated  and  give 


—yx}  —  C^   (10) 

where  Cv  Cv  Cs  are  constants  of  integration  (comp.  Art.  94) 
Hence,  if  the  external  forces  pass  through  a  fixed  point,  the 
angular  momentum  of  the  body  about  any  line  through  this 
point  is  constant  ;  if  there  are  no  external  forces,  the  angular 
momentum  is  constant  for  any  line  whatever.  This  is  the 
principle  of  the  conservation  of  angular  momentum. 

231.  Another  interpretation  can  be  given  to  these  equations 
As  shown  in  Arts.  88  and  91,  the  quantities  y'z  —  zy,  zx—xz 
xy—yx  can  be  regarded  as  sectorial  velocities.  Thus,  if  the 
radius  vector,  drawn  from  the  origin  to  the  particle  m,  be  pro 
jected  on  thejs'-plane,  y'z—  zy  is  twice  the  sectorial  velocity  of  this 
radius  vector  in  the  jj/^-plane,  \(ydz—  zdy)  being  the  elementary 
sector  described  in  the  element  of  time  dt.  Let  us  denote  by 
dSx  the  sum  of  all  these  elementary  sectors  for  the  various 
particles,  each  multiplied  by  the  mass  of  the  particle  ;  and 


232.]  GENERAL   PRINCIPLES. 

similarly  by  dSy,  dSz  the  corresponding  sums  of  the  projections 
on  the  other  co-ordinate  planes.  Then  the  equations  (10)  can 
be  written  in  the  form 

2SX=CV    2Sy=C2,    2SZ=C3.  (I  I) 

Hence  the  proposition  of  Art.  230  might  be  called  the  principle 
of  the  conservation  of  sectorial  velocities  ;  it  is  more  commonly 
called  the  principle  of  the  conservation  of  areas. 

The  equations  (11)  can  be  integrated  again  and  give,  if  the 
sectors  be  measured  from  the  positions  of  the  radii  vectores  at 
the  time  t=o, 


232.  If  the  radii  vectores  be  projected  on  any  plane  through 
the  origin  whose  normal  has  the  direction  cosines  a,  fi,  y,  the 
sum  of  the  elementary  sectors  described  in  this  plane,  each 
multiplied  by  the  mass,  will  be 


hence  S 

On  the  other  hand,  by  (10),  the  angular  momentum  of  the 
body  about  the  normal  of  this  plane  has  the  expression 

a  +  €<$  +  Csy,  as  it  must  be  equal  to  the  sum  of  the  pro- 
jections on  this  normal  of  the  angular  momenta  about  the 
axes  of  co-ordinates,  which  can  be  regarded  as  vectors  laid 
off  on  these  axes. 

Now  it  is  easy  to  see  that  this  angular  momentum  C^+CJS 
-\-C3y,  and  hence  the  quantity  5  at  a  given  time  /,  is  greatest 
for  the  diagonal  of  the  parallelepiped,  whose  edges  are  equal 
to  Clt  Cz,  C3  along  the  axes,  i.e.  for  the  normal  to  the  plane 

Cjx+C^y+C^  =  o.  (12) 

For,    the   direction    cosines    of    this    normal    are 


',  where  D=  V£\24-  C£+C£  ;  and  the  quan- 
tity C^a.  4-  C^ft  4-  QY  can  be  put  into  the  form 


128  KINETICS   OF   A   RIGID    BODY.  [233. 

where  the  quantity  in  parenthesis  is  the  cosine  of  the  angle 
between  the  directions  («',  /3',  7')  and  (a,  (3,  7),  and  is  therefore 
greatest  when  these  directions  coincide. 

The  plane  (12)  about  whose  normal  the  angular  momentum 
is  greatest,  and  by  projection  on  which  the  area  5  is  made 
greatest,  is  called  Laplace's  invariable  plane.  As  its  equation  is 
independent  of  /,  it  remains  fixed.  The  normal  of  this  plane  is 
sometimes  called  the  invariable  line  or  direction. 

233.  Let  us  now  return  to  the  general  case  of  the  motion  of 
a  rigid  body  acted  upon  by  any  forces  whatever. 

The  propositions  of  Arts.  226  and  229  together  establish  the 
so-called  principle  of  the  independence  of  the  motions  of  translation 
and  rotation.  In  studying  the  motion  of  a  rigid  body  it  is 
possible,  according  to  this  principle,  to  consider  separately  the 
motion  of  translation  of  the  centroid,  and  the  rotation  of  the 
body  about  the  centroid. 

By  Art.  226,  the  motion  of  the  centroid  is  the  same  as  that 
of  a  particle  of  mass  M  acted  upon  by  all  the  external  forces 
transferred  parallel  to  themselves  to  the  centroid.  As  the 
motion  of  a  particle  has  been  discussed  in  Chapter  V.,  nothing 
further  need  be  said  about  this  part  of  the  problem. 

By  Art.  229,  the  motion  of  the  body  about  the  centroid  is  the 
same  as  if  the  centroid  were  fixed.  The  problem  of  the  motion 
of  a  rigid  body  with  a  fixed  point  is  therefore  of  great  impor- 
tance ;  it  will  be  discussed  in  Section  IV.  The  more  simpl< 
special  case  of  a  rigid  body  with  a  fixed  axis  is  treated  in  Sec- 
tion III.  The  solution  of  both  these  problems  depends  on  the 
equations  (6)  or  (7). 

234.  In  d'Alembert's  equation  (3)  it  is  of  course  allowable  to 
substitute  for  the  virtual  displacements  %x,  8y,  &z  the  actual  dis- 
placements dx,  dy,  dz  of  the  particles  in  any  motion  of  a  fre< 
rigid  body,  since  these  actual  displacements  are  certainly  com- 
patible with  the  condition  of  rigidity.     The  equation  can  thei 
be  written 

(xdx  +ydy + zdz)  =  2  (Xdx  +Ydy  +  Zde).  (13) 


236.]  GENERAL   PRINCIPLES. 

The  left-hand  member  of  this  equation  evidently  represents 
the  differential  of  the  kinetic  energy 

r=%^2=%/^2+>2+^2)  (14) 

of  the  body,  while  the  right-hand  member  is  the  elementary 
work  of  the  external  forces.  Hence  equation  (13)  expresses 
the  principle  of  kinetic  energy  for  a  free  rigid  body,  viz.  the 
proposition  that,  for  any  infinitesimal  displacement  of  the  body, 
the  increase  of  the  kinetic  energy  is  equal  to  the  sum  of  the  works 
done  by  all  the  external  forces. 

235.  By  introducing  the  co-ordinates  of  the  centroid,  i.e.  by 
putting  x=x+%y  y=y^-r),  £=#  +  £,  as  in  Art.  228,  the  expres- 
sion for  the  kinetic  energy  assumes  the  form  (since  ^m^=o, 


(is) 

where  v  is  the  velocity  of  the  centroid  and  u  the  relative  veloc- 
ity of  any  particle  m  with  respect  to  the  centroid. 

Thus,  it  appears  that  the  kinetic  energy  of  a  free  rigid  body 
consists  of  two  parts,  one  of  which  is  the  kinetic  energy  of  the  cen- 
troid (the  whole  mass  being  regarded  as  concentrated  at  this 
point),  while  the  other  may  be  called  the  relative  kinetic  energy 
with  respect  to  the  centroid. 

236.  By  the  same  substitution  the  right-hand  member  of 
equation  (13),  i.e.  the  elementary  work  ^(Xdx+  Ydy  +  Zdz), 
resolves  itself  into  the  two  parts 

(dx  $X+  dy  2  F-f  dz  2Z)  +  2  (Xd%  4-  Ydri  +  Zd£). 

The  first  parenthesis  contains  the  work  that  would  be  done  by 
all  the  external  forces  if  they  were  applied  at  the  centroid ;  it  is 
therefore  equal  to  the  kinetic  energy  of  the  centroid,  that  is  to 
mfyMv*).  The  equation  of  kinetic  energy  (13)  reduces,  there- 
fore, to  the  following : 

(16) 

PART  m-9  ^- 


KINETICS   OF   A   RIGID   BODY.  [237 

in  other  words,  the  principle  of  kinetic  energy  holds  for  the  rela- 
tive motion  with  respect  to  the  centroid. 

237.  Impulses.  The  equations  determining  the  effect  of  a 
system  of  impulses  (see  Arts.  2-5)  on  a  rigid  body  are  readily 
obtained  from  the  general  equations  of  motion  (4)  and  (6). 
We  shall  denote  the  impulse  of  a  force  F  by  F.  It  will  be 
remembered  that  the  impulse  F  of  a  force  F  is  its  time  in- 
tegral ;  i.e. 


We  confine  ourselves  to  the  case  when  t1  —  t  is  very  small  and  F 
very  large,  in  which  case  the  action  of  the  impulsive  force  F  is 
measured  by  its  impulse  F. 

If  all  the  forces  acting  on  a  rigid  body  are  of  this  nature,  and 
the  impulses  of  X,  Y,  Z  during  the  short  interval  t'  —  t  be 
denoted  by  X,  Y,  Z,  the  integration  of  the  equations  (4)  from 
t=t  to  t=tf  gives 

^m(x'-x)  =  ?.X,  2w(>'-»  =  2F,  2f»(*'-*)  =  2Z,      (17) 

where  x,  j,  z  denote  the  velocities  of  the  particle  m  at  the  time 
/  just  before  the  impulse,  and  x\  y\  z'  those  at  the  time  /'  just 
after  the  action  of  the  impulse. 
Similarly  the  equations  (6)  give 


l-x)-x(z1-z)]  =  ^(zX-xZ)t  (18) 

x(y<  -  y)  -y(x'  -x)}  =  2(*  Y-yX). 

238.  In  determining  the  effect  on  a  rigid  body  of  a  system 
of  such  impulses,  any  ordinary  forces  acting  on  the  body  at  the 
same  time  are  neglected  because  the  changes  of  velocity  pro- 
duced by  them  during  the  very  short  time  r  are  small  in  com- 
parison with  the  changes  of  velocity  x1  —  x,  y1  —y,  z'  —z  produced 
by  the  impulses.  For  the  mathematical  treatment  it  is  generally 

most  convenient  to  define  the  impulse  F  of  an  impulsive  force 

Jf 
Fdt  when  /'  —  /  approaches  o  and 
*- 


238.]  GENERAL   PRINCIPLES.  !3I 

F  approaches  oc  (Art.  5)  ;  in  this  case  it  is  strictly  true  that  the 
effect  of  ordinary  forces  can  be  neglected  when  impulsive  forces 
act  on  the  body. 

If  the  rigid  body  be  originally  at  rest,  it  will  be  convenient 
to  denote  by  x,  y,  z  the  components  of  the  velocity  of  the  par- 
ticle m  just  after  the  action  of  the  impulses.  We  may  also 
denote  by  R  the  resultant  of  all  the  impulses,  by  H  the  result- 
ant impulsive  couple  for  the  reduction  to  the  origin  of  co- 
ordinates, and  mark  the  components  of  R  and  H  by  subscripts, 
as  in  the  case  of  forces.  With  these  notations  the  effect  of  a 
system  of  impulses  on  a  body  at  rest  is  given  by  the  equations 

(19) 
Hg.     (20) 


In  the  equations  (19)  we  have,  of  course,  ^mx=Mx, 
z=Mz,  where  x,  y,  ~z  are  the  components  of  the  velocity  of 
the  centroid,  and  M  is  the  mass  of  the  body  ;  i.e.  the  momentum 
of  the  centroid  is  equal  to  the  resultant  impulse.  The  meaning 
of  the  equations  (20)  can  be  stated  by  saying  that  the  angular 
momentum  of  the  body  about  any  axis  is  equal  to  the  moment  of 
all  the  impttlses  about  the  same  axis. 


KINETICS   OF   A   RIGID    BODY.  [239. 

II.    Moments  of  Inertia  and  Principal  Axes. 

-«• 

I.    INTRODUCTION. 

239.  As  will  be  shown  in  Sections  III.  and  IV.,  the  rotation 
of  a  rigid  body  about  any  axis  depends  not  only  on  the  forces 
acting  on  the  body,  but  also  on  the  way  in  which  the  mass 
is  distributed  throughout  the  body.     This  distribution  of  mass 
is  characterized  by  the  position  of  the  centroid  and  by  that  of 
certain  lines  in  the  body  called  principal  axes. 

It  has  been  shown  in  Part  II.,  Art.  13,  that  the  centroid  of  a 
mass  is  found  by  determining  the  moments,  or  more  precisely, 
the  moments  of  the  first  order,  ^mx,  ^my,  *Zmz,  of  the  mass  with 
respect  to  the  co-ordinate  planes,  i.e.  the  sums  of  all  mass- 
particles  m  each  multiplied  by  its  distance  from  the  co-ordinate 
plane. 

The  principal  axes  of  a  mass  or  body  can  be  found  by  deter- 
mining .the  moments  of  the  second  order,  ^mx2,  ^my2,  ^mx*, 
^myz,  ^mzx,  ^mxy  of  the  mass  with  respect  to  the  same 
planes.  We  proceed,  therefore,  to  study  the  theory  of  such 
moments. 

240.  If  in  a  rigid  body  the  mass  m  of  each  particle  be  multi- 
plied by  the  square  of  its  distance  r  from  a  given  point,  plane, 
or  line,  the  sum 

2  mr*  =  m^rf  +  m^rf  -f  -  •  •, 

extended  over  the  whole  body,  is  called  the  quadratic  moment, 
or,  more  commonly,  the  moment  of  inertia  of  the  body  for  that 
point,  plane,  or  line. 

If  the  body  is  not  composed  of  discrete  particles,  but  forms 
a  continuous  mass  of  one,  two,  or  three  dimensions,  this  mass 
can  be  resolved  into  elements  of  mass  dm,  and  the  sum 
becomes  a  single,  double,  or  triple  integral  (r*dm. 


243-]  MOMENTS   OF   INERTIA. 


Expressions  of  the  form  ^mr^r^  or  r^dm,  where  rv  r%  are 
the  distances  of  m  or  of  dm  from  two  planes  (usually  at  right 
angles),  are  called  moments  of  deviation  or  products  of  inertia. 

241.  The  determination  of  the  moment  of  inertia  of  a  con- 
tinuous mass   is  a  mere  problem  of  integration  ;  the  methods 
are  similar  to  those  for  finding  the  moments  of  mass  of  the  first 
order  required  for  determining  centroids  (see  Part  II.,  Chapter 
III.),  the  only  difference  being  that  each  element  of  mass  must 
be  multiplied  by  the  square,  instead  of  the  first  power,  of  the 
distance. 

A  moment  of  inertia  is  not  a  directed  quantity  ;  it  is  not 
a  vector,  but  a  scalar  ;  indeed,  it  is  a  positive  quantity,  provided 
the  masses  are  all  positive,  as  we  shall  here  assume. 

The  moment  of  inertia  of  any  number  of  bodies  or  masses 
for  any  given  point,  plane,  or  line  is  obviously  the  sum  of  the 
moments  of  inertia  of  the  separate  bodies  or  masses  for  the 
same  point,  plane,  or  line. 

242.  The  moment  of  inertia  ^mr*  of  any  body  whose  mass 
is  M=  2<m  can  always  be  expressed  in  the  form 


where  r0  is  a  length  called  the  radius  of  inertia,  arm  of  inertia, 
or  radius  of  gyration.  This  length  rQ  is  evidently  a  kind  of 
average  value  of  the  distances  r,  its  value  being  intermediate 
between  the  greatest  r1  and  least  r"  of  these  distances  r.  For 
we  have  2,mr'2  >2mr2  >2mr"2,  or,  since 


243.  As  an  example,  let  us  determine  the  moment  of  inertia 
of  a  homogeneous  rectilinear  segment  (straight  rod  or  wire  of 
constant  cross-section  and  density)  for  its  middle  point  (or, 
what  amounts  to  the  same  thing,  for  a  line  or  plane  through 
this  point  at  right  angles  to  the  segment). 


134  KINETICS   OF   A   RIGID    BODY.  [244. 

Let  2 1  be  the  length  of  the  rod  (Fig.  28),  O  its  middle  point, 
p  its  density  (i.e.  the  mass  of  unit  length),  x  the  distance  OP 


A  O  P  B 

Fig.  28. 

of   any   element   dm  =  pdx  from   the   middle  point.     Then  we 
have,  for  the  moment  of  inertia  /, 


and  for  the  radius  of  inertia  r0,  since  the  whole  mass  is  M=  2  pi, 

r*=*-=\l\ 

244.  Exercises. 

Determine  the  radius  of  inertia  in  the  following  cases.     When  noth- 
ing is  said  to  the  contrary,  the  masses  are  supposed  to  be  homogeneous. 

(1)  Segment  of  straight  line  of  length  /,  for  a  perpendicular  through 
one  end. 

(2)  Rectangular  area  of  length  /  and  width  h  :   (a)  for  the  side  h ; 
(£)  for  the  .side  /;  (c)  for  a  line  through  the  centroid  parallel  to  the 
side  h ;  (*/)  for  a  line  through  the  centroid  parallel  to  the  side  /. 

(3)  Triangular  area  of  base  b  and  height  h,  for  a  line  through  the 
vertex  parallel  to  the  base. 

(4)  Square  of  side  a,  for  a  diagonal. 

(5)  Regular  hexagon,  for  a  diagonal. 

(6)  Right  cylinder  or  prism  of  height  2  ht   for  the  plane  bisecting 
the  height  at  right  angles. 

(7)  Segment  of  straight  line  of  length  /,  for  one  end,  when  the  density 
is  proportional  to  the  nih  power  of  the  distance  from  this  end.     Deduce 
from  this:    (a)   the  result  of  Ex.   (i)  ;   (b)  that  of  Ex.   (3)  ;   (c)   the 
radius  of  inertia  of  a  homogeneous  pyramid  or  cone  (right  or  oblique) 
of  height  h,  for  a  plane  through  the  vertex  parallel  to  the  base. 

(8)  Circular  area  (plate,  disc,  lamina)  of  radius  a,  for  any  diameter. 

(9)  Circular  line  (wire)  of  radius  a,  for  a  diameter. 


246.] 


MOMENTS   OF   INERTIA. 


135 


(10)   Solid  sphere,  for  a  diametral  plane. 

(n)   Solid  ellipsoid,  for  the  three  principal  planes. 

(12)  Area  of  ring  bounded  by  concentric  circles  of  radii  aly  az,  for  a 
diameter. 

(13)  Area   of  the    cross-section   of  a    JL-iron  :    (a)   for  its  line  of 
symmetry;   (b)  for  its  base.     (Dimensions  as  in  Fig.  8,  Part  II.,  p.  18.) 

(14)  A  rectangular  door  of  width  b  and  height  h  has  a  thickness  8  to 
a  distance  a  from  the  edges,  while  the  rectangular  panel  (whose  dimen- 
sions are  b  —  2  a,  h  —  2  a)  has  half  this  thickness.     Find  the  moment 
of  inertia  for  a  line  through  the  centroid  parallel  to  the  side  b. 

245.  The  moment  of  inertia  of  any  mass  M  for  a  point  can 
easily  be  found  if  the  moments  of  inertia  of  the  same  mass 
are  known  for  any  line  passing  through  the 
point,  and  for  the  plane  through  the  point 
perpendicular  to  the  line.  Let  O  (Fig.  29) 
be  the  point,  /  tbe  line,  TT  the  plane ;  r,  q,  p 
the  perpendicular  distances  of  any  particle  of 
mass  m  from  O,  /,  TT,  respectively.  Then 
we  have,  evidently,  r2=g>2-\-fl2.  Hence,  mul- 
tiplying by  m,  and  summing  over  the  whole 
mass  M, 


or,  putting 

are  the  radii  of  inertia  for  O,  /,  TT, 


Fig.  29. 


,  where  r0, 


(i) 


246.  The  moment  of  inertia  of  any  mass  M  for  a  line  is 
equal  to  the  sum  of  the  moments  of  inertia  of  the  same  mass 
for  any  two  rectangular  planes  passing  through  the  line.  Thus, 
in  particular,  the  moment  of  inertia  for  the  axis  of  x  in  a 
rectangular  system  of  co-ordinates  is  equal  to  the  sum  of  the 
moments  of  inertia  for  the  sar-plane  and  ;rj/-plane.  This  fol- 
lows at  once  by  considering  that  the  square  of  the  distance 
of  any  point  from  the  line  is  equal  to  the  sum  of  the  squares 


136 


KINETICS    OF   A   RIGID    BODY. 


[247. 


of  the  distances  of  the  same  point  from  the  two  planes.  Thus, 
if  q  be  the  distance  of  any  point  (x,  y,  z)  from  the  axis  of  xy 
we  have  <=L+&  whence 


247.  It  follows,  from   the   last   article,  that   the  moment  of 
inertia  lx   of  a  plane  area,  for  any    line  perpendicular  to   its 
plane,  is 

/.=/,+/* 

if  fy,  Iz  are  the  moments  of  inertia  of  the  area  for  any  two 
rectangular  lines  in  the  plane  through  the  foot  of  the  perpen- 
dicular line. 

248.  The  problem  of  finding  the  moment  of  inertia  of  a  given 
mass  for  a  line  1',  when  it  is  known  for  a  parallel  line  1,  is  of 

great  importance. 


Let  Hmq2'  be  the  moment  of  inertia  of  the 
given  mass  for  the  line  /  (Fig.  30),  ^mql2> 
that  for  a  parallel  line  /'  at  the  distance  d 
from  /.  The  distances  q,  q1  of  any  particle 
m  from  /,  /'  form  with  d  a  triangle  which 
gives  the  relation 

S  (q,  d). 


Fig.  30.  f* 

Multiplying  by  m,  and  summing  over  the  whole  mass  M,  we 
find 

(q,  d). 


Now  the  figure  shows  that  the  product  qcos(q,d)  in  the 
last  term  is  the  distance  /  of  the  particle  m  from  a  plane  through 
/  at  right  angles  to  the  plane  determined  by  /  and  /'.  We  have, 
therefore, 

(2) 


where  the  last  term  contains  the  moment  of   the  first  order 
=  Mp  of  the  given  mass  M  for  the  plane  just  me'ntioned. 


251.]  MOMENTS    OF   INERTIA.  137 

If,  in  particular,  this  plane  contains  the  centroid  G  of   the 
mass  Mt  we  have  2w/  =  o,  so  that  the  formula  reduces  to 

^mq^  =  2mg2  +  Md*.  (3) 

Introducing  the  radii  of  inertia  ^0',  ^0,  this  can  be  written 

(3') 


249.    Similar  considerations  hold  for  the  moments  of  inertia 
^mp®   with    respect  to   two  parallel  planes  TT,  TT'  at  the 
distance  d  from  each  other.     We  have,  in  this  case,  /'=/  —  d\ 


,  (4) 

and  if  the  plane  TT  contain  the  centroid  G, 

(5) 


250.  Of  special  importance  is  the  case  in  which  one  of  the 
lines  (or  planes),  say  /  (TT},  contains  the  centroid.     The  formulae 
(3)>  (3;)>  and  (5)  hold  in  this  case  ;  and  if  we  agree  to  designate 
any  line  (plane)   passing  through  the  centroid  as  a  centroidal 
line  (plane),  our  proposition  can  be  expressed  as  follows  :   The 
moment  of  inertia  for  any  line  (plane]  is  found  from  the  moment 
of  inertia  for  the  parallel  centroidal  line  (plane]  by  adding  to 
the  latter  the  product  Md2  of  the  whole  mass  into  the  square  of 
the  distance  of  the  lines  (planes]. 

It  will   be   noticed   that    of    all    parallel   lines    (planes)   the 
centroidal  line  (plane)  has  the  least  moment  of  inertia. 

251.  Exercises. 

Determine  the  radius  of  inertia  of  the  following  homogeneous  masses  : 

(1)  Rectangular  plate  of  length  /and  width  h,  for  a  centroidal  line 
perpendicular  to  its  plane. 

(2)  Area  of  equilateral  triangle  of  side  a  :   (a)  for  a  centroidal  line 
parallel  to  the  base  ;   (b)  for  an  altitude  ;   (c)  for  a  centroidal  line  per- 
pendicular to  its  plane. 

(3)  Circular  disc  of  radius  a:   (a)   for  a  tangent;   (d)   for  a  line 
through  the  centre  perpendicular  to  the  plane  of  the  disc  ;   (c)  for  a 
perpendicular  to  its  plane  through  a  point  in  the  circumference. 


I38  KINETICS   OF   A   RIGID   BODY.  [252. 

(4)  Solid  sphere,  for  a  diameter. 

(5)  Area  of  ring  bounded  by  concentric  circles  of  radii  alt  a2,  for 
a  line  through  the  centre  perpendicular  to  the  plane  of  the  ring.     For 
a  ring  whose  thickness  a2  —  a±  is  infinitesimal,  the  result  can  also  be 
obtained  by  differentiation  from  Ex.  (3)  (b). 

(6)  Spherical  shell  of  infinitesimal  thickness,  for  a  diameter. 

(7)  Right  circular  cylinder,  of  radius  a  and   height  2h  :   (a)  for 
its  axis ;   (b)  for  a  generating  line  ;   (c)  for  a  centroidal  line  in  the  mid- 
dle cross-section. 

(8)  Prove  that,  in  a  right  prism  or  cylinder  of  any  cross-section, 
we  have  <?2  =  q*  +  ^c2,  where  q  is  the  radius  of  inertia  of  the  prism  or 
cylinder  for  a  line  bisecting  the  axis  at  right  angles,  qa  the  radius  of 
inertia  of  the  axis,  qc  that  of  the  middle  cross-section,  for  the  same  line. 

(9)  Area  of  ellipse  :   (a}  for  the   major   axis ;    (b)   for  the  minor 
axis  ;   (c)  for  the  perpendicular  to  its  plane  through  the  centre. 

(10)  Solid  ellipsoid,  for  each  of  the  three  axes. 

(n)  Area  of  the  cross-section  of  a  1-iron,  for  a  centroidal  line  par- 
allel to  the  flange.  (Compare  Art.  244,  Ex.  (13).) 

(12)  Area   of  the   cross-section   of  a   symmetrical   double  T-iron, 
width  of  flanges  b,  thickness  of  flanges  8,  height  of  web  h,  thickness 

•  of  web  28;  for  the  two  axes  of  symmetry,  and  for  a  centroidal  line  per- 
pendicular to  its  plane. 

(13)  Wire  bent  into  an  equilateral  triangle  of  side  a,  for  a  centroidal 
line  at  right  angles  to  the  plane  of  the  triangle. 

252.  Bouth's  Rule.  In  the  case  of  homogeneous  masses  with 
axes  of  symmetry,  the  radius  of  inertia  for  an  axis  of  symmetry 
can  readily  be  derived  by  the  following  mnemonical  rule :  The 
square  of  the  radius  of  inertia  is  ^,  \,  or  \  of  the  sum  of  the 
squares  of  the  perpendicular  semi-axes,  according  as  the  mass  is 
rectangular,  elliptic,  or  ellipsoidal. 

The  proof  rests  on  the  following  typical  cases  which  are 
easily  proved  directly  (comp.  Art.  251,  Ex.  (i),  (9),  (10)) : 

(i)  Rectangular  area  whose  sides  are  2  a,  2b,  for  a  centroidal 
line  perpendicular  to  its  plane  :  g2  =  ^ 


255-]  ELLIPSOIDS   OF  INERTIA. 

(2)  Elliptic  area  whose  axes  are  2  a,  2b,  for  a  centroidal  line 
perpendicular  to  its  plane  :  q>2=^(a2-\-&2). 

(3)  Solid  ellipsoid  whose  axes  are  2a,  2b,  2c,  for  its  axes  : 


A  large  number  of  special  cases  can  be  brought  under  this 
rule,  as  will  be  seen  from  the  following  exercises.  It  should  be 
remembered  that  the  radius  of  inertia  of  a  homogeneous  right 
prism  or  cylinder  for  its  axis  is  the  same  as  that  of  its  cross- 
section. 

253.  Exercises.     Apply  Routh's  rule  to  find  the  radius  of  inertia  in 
the  following  cases  : 

(  i  )  Solid  sphere  of  radius  a,  for  a  diameter. 

(2)  Right  circular  cylinder,  for  its  axis. 

(3)  Thin  straight  rod  of  length  2  a,  for  a  perpendicular  through  its 
middle  point. 

(4)  Rectangular  disc  whose  sides  are  2  a,  2  b,  for  a  line  in  its  plane 
bisecting  the  sides  2  a. 

(5)  Circular  disc,  for  a  diameter. 

2.    ELLIPSOIDS    OF    INERTIA. 

254.  The  moments  of  inertia  of  a  given  mass  for  the  different 
lines  of  space  are  not  independent  of  each   other.       Several 
examples  of  this  have  already  been  given.     It  has  been  shown, 
in  particular  (Art.  248),  that  if  the  moment  of  inertia  be  known 
for  any  line,  it  can  be  found  for  any  parallel  line.     It  follows 
that  if  the  moments  be  known  for  all  lines  through  any  given 
point,  the  moments  for  all  lines  of  space  can  be  found.     We 
now  proceed  to  study  the  relations  between  the  moments   of 
inertia  for  all  the  lines  passing  through  any  given  point  O. 

255.  It  will  here  be  convenient  to  refer  the  given  mass  M  to 
a  rectangular  system  of  co-ordinates  with  the  origin  at  the  point 
O.     Let  x,  y,  z  be  the  co-ordinates  of  any  particle  m  of  the 


140 


KINETICS    OF   A   RIGID    BODY. 


[256, 


mass  ;  and  let  us  denote  by  A,  B,  C  the  moments  of  inertia  of 
M  for  the  axes  of  xy  y,  z\  by  A\  B\  O  those  for  the  planes  yzr 
zx,  xy\\>yD)EyF  the  products  of  inertia  (Art.  240)  for  the 
co-ordinate  planes  ;  i.e.  let  us  put  : 


B'  =  2mjP,          E=^mzx,  (6) 

C  =  2m  (x*  +y2),  C  =  ^mz*,  F=  ^mxy. 

256.    These  nine  quantities  are  not  independent  of  each  other. 
We  have  evidently 

A=B'  +  C',  B= 

hence,  solving  for  A',  B\  C', 


The  moment  of  inertia  for  the  origin  O  is 


).        (7) 

257.  The  moment  of  inertia  7  for  any  line  through  O  can  be 
expressed  by  means  of  the  six  quantities  A,  B,  C,  D,  E,  F\  and 
the  moment  of  inertia  /'  for  any  plane  through  O  can  be 
expressed  by  means  of  A',  B\  C,  D,  E,  F. 

Let  TT  (Fig.  31)  be  any  plane  passing  through  O  ;  /its  normal  ; 
a,  j3,  7  the  direction  cosines  of  /;  and,  as  before  (Art.  245),  p, 

q,  r  the  distances  of  any  point  (x,y,  s) 
of  the  given  mass  from  TT,  /,  and  O, 
respectively.  Then,  projecting  the 
closed  polygon  formed  by  r,  x,  y,  z 
on  the  line  /,  we  have 


Fig.  31. 


hence,  squaring,  multiplying  by 
m,  and  summing  over  the  whole 
mass,  we  find 


or,  with  the  notations  (6), 

r  =  A  'a2  +  £'/32  +  C  V  4-  2  D/3y  +  2  Eja  +  2  Fa/3. 


(8) 


259-]  ELLIPSOIDS   OF   INERTIA.  !4I 

Thus  the  moment  of  inertia  for  any  plane  through  the  origin  is 
expressed  as  a  homogeneous  quadratic  function  of  the  direction 
cosines  of  the  normal  of  the  plane. 


258.   The  moment  of  inertia  f=2mq2  for  the  line  /  can  now 
be  found  from  equation  (i),  Art.  245,  by  substituting  for 
and  2;/z/2  their  values  from  (7)  and  (8)  : 


or,  snce  «  +  /    +  7=  i, 


hence,  finally,  applying  the  relations  of  Art.  256, 

(9) 


The  moment  of  inertia  for  any  line  through  the  origin  is, 
therefore,  also  a  homogeneous  quadratic  function  of  the~direction 
cosines  of  the  line. 

259.  These  results  suggest  a  geometrical  interpretation.  Im- 
agine an  arbitrary  length  OP=p  laid  off  from  the  origin  O  on 
the  line  /  whose  direction  cosines  are  a,  ft,  7  ;  the  co-ordinates 
of  the  extremity  •  P  of  this  length  will  be  x  =  pa,  y  =  p/3,  z  =  py. 
Now,  if  equation  (9)  be  multiplied  by  p2,  it  assumes  the  form 


which  represents  a  quadratic  surface  provided  that  p  be  so 
selected  for  the  different  lines  through  O  as  to  make  p2!  con- 
stant, say/32/=A;2  Hence,  if  on  every  line  1  through  the  origin 
a  length  OP  =  /o  =  /<:/VT  be  laid  off,  i.e.  a  length  inversely  pro- 
portional to  the  square  root  of  the  moment  of  inertia  I  for  this 
line  1,  the  points  P  will  lie  on  the  quadric  surface 


I42  KINETICS   OF   A   RIGID   BODY.  [260. 

The  constant  /c2  may  be  selected  arbitrarily  ;  to  preserve  the 
homogeneity  of  the  equation  it  will  be  convenient  to  put  it  into 
the  form  /c2  =  J/e4,  where  e  is  still  arbitrary. 

260.  As  moments  of  inertia  are  essentially  positive  quantities, 
the  radii  vectores  of  the  surface 


(10) 

are  all  real,  and  the  surface  is  an  ellipsoid.  It  is  called  the 
ellipsoid  of  inertia,  or  the  momental  ellipsoid,  of  the  point  O. 
This  point  O  is  the  centre  ;  the  axes  of  the  ellipsoid  are  called 
the  principal  axes  at  the  point  O  ;  and  the  moments  of  inertia 
for  these  axes  are  called  the  principal  moments  of  inertia  at  the 
point  O.  Among  these  there  will  evidently  be  the  greatest  and 
least  of  all  the  moments  of  the  point  O,  the  greatest  moment 
corresponding  to  the  shortest,  the  least  to  the  longest  axis  of 
the  ellipsoid. 

It  may  be  observed  that,  owing  to  the  relations  of  Art.  256, 
which  show  that  the  sum  of  any  two  of  the  quantities  A,  B,  C 
is  always  greater  than  the  third,  not  every  ellipsoid  can  be 
regarded  as  the  momental  ellipsoid  of  some  mass.  An  ellipsoid 
can  be  a  momental  ellipsoid  only  when  a  triangle  can  be  con- 
structed of  its  semi-axes. 

261.  If  the  axes  of  the  ellipsoid  (10)  be  taken  as  axes  of 
co-ordinates,  the  equation  assumes  the  form 


where  Jlt  72,  73  are  the  principal  moments  at  the  point  O. 

By  Art.  259  we  have  ?  =K*/f=  M£J  'I  ;  hence  I=M*/f?.  If, 
therefore,  equation  (11)  be  divided  by/:)2,  the  following  simple 
expression  is  obtained  for  finding  the  moment  of  inertia,  /,  for 
a  line  whose  direction  cosines  referred  to  the  principal  axes 
are  «,  /3,  7, 

"-      (12) 


263.]  ELLIPSOIDS   OF   INERTIA. 

262.  To  make  use  of  this  form  for  /,  the  principal  axes  at  the  point 
O,  i.e.  the  axes  of  the  momental  ellipsoid  (10),  must  be  known.  The 
determination  of  the  axes  of  an  ellipsoid  whose  equation  referred  to  the 
centre  is  given  is  a  well-known  problem  of  analytic  geometry.  It  can 
be  solved  by  considering  that  the  semi-axes  are  those  radii  vectores  of 
the  surface  that  are  normal  to  it.  The  direction  cosines  of  the  normal 
of  any  surface  F(x,  y,  z)  =  o  are  proportional  to  the  partial  derivatives 
dF/dx,  dF/dy,  dF/dz.  If,  therefore,  the  radius  vector  p  is  a  semi- 
axis,  its  direction-cosines  a,  ft,  y  must  be  proportional  to  the  partial 
derivatives  of  (10)  ;  i.e.  we  must  have 

Cz 


or  dividing  the  numerators  by  p, 

Aa-F(3—Ey  =  -  Fa  +  Bft  -  Dy  __  -  Ea  -  Dft  +  Cy 
a  (3  y 

Denoting  the  common  value  of  these  fractions  by  /,  we  have 

al=  Aa  —Fft  -  Ey,  ftl=  -  Fa  +  Bft  -  Dy,  y/=  -  Ea  -  Dft  +  Cy  ; 

multiplying  these  equations  by  a,  ft,  y,  and  adding,  we  find 

/  =  A  a2  +  B^  +  Cy2  -  2  Dfty  —  2  Eya  —  2  Faft^  . 

which,  compared  with  (9),  shows  that  /  is  the  moment  of  inertia  for 
the  axis  (a,  ft,  y)  .  To  obtain  it  in  function  of  A,  B,  C,  D,  E,  F,  we 
write  the  preceding  three  equations  in  the  form 

(S-A)a  +  Fft+  Ey=o, 

£>y  =  o,  (13) 


whence,  eliminating  a,  ft,  y,  we  find  /  determined  by  the  cubic  equation 
I  -A,         F,         E 


F,  I-B,         D 


=  o.  (14) 


E,         D,  I-  C 

The  roots  of  this  cubic  are  the  three  principal  moments  /i,  /2,  /3  of  the 
point  O.  The  direction-cosines  of  the  principal  axes  are  then  found  by 
substituting  successively  Il9  72,  73  in  (13)  and  solving  for  a,  ft,  y. 

263.    The    geometrical    representation    of    the    moments    of 
inertia  for  all  lines  passing  through  a  point  by  means  of  the 


144  KINETICS   OF   A   RIGID    BODY.  [264. 

radii  vectores  of  the  momental  ellipsoid  at  the  point,  gives  at 
once  a  number  of  propositions  about  these  moments.  It  is 
only  necessary  to  interpret  properly  the  geometrical  properties 
of  the  ellipsoid.  Thus,  it  is  known  that  the  sum  of  the  squares 
of  the  reciprocals  of  any  three  rectangular  semi-diameters  of  an 
ellipsoid  is  constant.  It  follows  that  the  sum  of  the  three 
moments  of  inertia  for  any  three  rectangular  lines  passing 
through  the  same  point  has  a  constant  value. 

In  general,  the  three  principal  moments  of  inertia  fv  72,  73 
at  a  point  O  are  different.  If,  however,  two  of  them  are  equal, 
say  /2  =  /3,  the  momental  ellipsoid  becomes  an  ellipsoid  of 
revolution  about  the  third,  flt  as  axis  ;  and  it  follows  that  the 
moments  of  inertia  for  all  lines  through  O  lying  in  the  plane 
of  the  two  equal  axes  are  equal. 

If  1^  =  1^  =  1^  the  ellipsoid  becomes  a  sphere,  and  the  mo- 
ments of  inertia  are  the  same  for  all  lines  passing  through  O. 

264.  If  the  equation  of  the  momental  ellipsoid  at  a  point  O 
be  of  the  form  Ax*  +  B}P+Cz*  —  2Dyz  =  J/e4;  i.e.  if  the  two  con- 
ditions 

E  =  ^mzx  =  o,  F  =  ^mxy  —  o 


be  fulfilled,  the  axis  of  x  coincides  with  one  of  the  three  axes 
of  the  ellipsoid,  the  surface  being  symmetrical  with  respect  to 
the  ^-plane.  Hence,  if  the  conditions  E  =  o,  F  =  o  are  satisfied, 
the  axis  of  K  is  a  principal  axis  at  the  origin.  The  converse  is 
evidently  also  true  ;  i.e.  if  a  line  is  a  principal  axis  at  one  of 
its  points,  then,  taking  this  point  as  origin  and  the  line  as  axis 
of  xt  the  conditions  ^m^  =  o,  ^mxy  =  o  must  be  satisfied. 

It  is  easy  to  see  that  if  a  line  be  a  principal  axis  at  one  of  its 
points,  say  O,  it  wjll  in  general  not  be  a  principal  axis  at  any 
other  one  of  its  points.  For,  taking  the  line  as  axis  of  x  and 

0  as  origin,  we  have  ^mzx=ot  ^mxy  =  o.     If  now  for  a  point 

01  on  this  line  at  the  distance  a  from  O  the  line  is  likewise 
a  principal  axis,  the  conditions 


266.]  ELLIPSOIDS   OF   INERTIA. 

must  be  fulfilled.     These  reduce  to 


and  show  that  the  line  must  pass  through  the  centroid.  And 
as  for  a  centroidal  line  these  conditions  are  satisfied  indepen- 
dently of  the  value  of  a,  it  appears  that  a  centroidal  principal 
axis  is  principal  axis  at  every  one  of  its  points.  Hence  a  line 
cannot  be  principal  axis  at  more  than  one  of  its  points  unless  it 
pass  through  the  centroid ;  in  the  latter  case  it  is  principal  axis 
at  every  one  of  its  points. 

265.  All  those  lines  passing  through  a  given  point  O  for 
which  the  moments  of  inertia  have  the  same  value  7  can  be 
shown  to  form  a  cone  of  the  second  order  whose  principal 
diameters  coincide  with  the  axes  of  the  momental  ellipsoid 
at  O.  This  cone  is  called  an  equimomental  cone.  Its  equation 
is  obtained  by  regarding  7  as  constant  in  equation  (12)  and 
introducing  rectangular  co-ordinates.  Multiplying  (12)  by 
;=  i,  we  find 


and  multiplying   by  p2,   we  obtain  the  equation   of   the  equi- 
momental cone  in  the  form 

266.  A  slightly  different  form  of  the  equations  (n),  (12),  (15) 
is  often  more  convenient ;  it  is  obtained  by  introducing  the 
three  principal  radii  of  •  inertia  q^  q^  qz  defined  by  the  relations 


The  equation  (11)  of  the  momental  ellipsoid  at  the  point  O  then 
assumes  the  form 


The  expression  of  the  radius  of  inertia  q  for  any  line  (a,  0,  7) 
through  O  becomes 


PART   III  —  10 


I46  KINETICS   OF   A   RIGID    BODY.  [267. 

Dividing  (nf)  by  the  square  of  the  radius  vector,  /a2,  and  com- 
paring with  (12'),  we  find 

e2          e2 
?=->/>=->  (16) 

as  is  otherwise  apparent  from  the  fundamental  property  of  the 
momental  ellipsoid  (Art.  259). 

The  equation  of  the  sphere  of  radius  q  described  about  O  as 
centre,  ^2+j2+^2=^2,  together  with  (iif),  represents  the  curve 
of  intersection  of  the  ellipsoid  with  the  sphere.  Through  this 
sphero-conic  passes  the  equimomental  cone,  all  of  whose  lines 
have  the  moment  of  inertia  I=Mq\  Hence,  the  equation  of 
this  cone  can  be  written  in  the  form 


267.  If  we  assume  II  >  72  >  73,  and  hence  q\>q^>  q&  q  must 
be  <  e2/^  and  >e2/^1.     As  long  as  q  is  less  than  the  middle 
semi-axis  e2/^2  of  the  ellipsoid,  the  axis  of  the  cone  coincides 
with  the  axis  of  x,  but  when  q>^/q^  the  axis  of  z  is  the  axis 
of  the  cone.     For  £  =  e2/^2  the  cone  degenerates  into  the  pair 
of  planes  (q\—qf)x'L—(q^—qf]z^=Q.     These  are  the  planes  of 
the  central  circular  (or  cyclic)   sections  of  the  ellipsoid  ;  they 
divide  the  ellipsoid  into  four  wedges,  of  which  one  pair  contains 
all  the  equimomental  cones  whose  axes  coincide  with  the  great- 
est axis  of  the  ellipsoid,  while  the  other  pair  contains  all  those 
whose  axes  lie  along  the  least  axis  of  the  ellipsoid. 

268.  There  is  another  ellipsoid  closely  connected   with  the 
theory  of    principal  axes  ;    it  is   obtained  from  the  momental 
ellipsoid  by  the  process  of  reciprocation. 

About  any  point  O  (Fig.  32)  taken  as  centre  let  us  describe 
a  sphere  of  radius  e,  and  construct  for  every  point  P  its 
polar  plane  TT  with  regard  to  the  sphere.  If  P  describe 
any  surface,  the  plane  TT  will  envelop  another  surface-.  which  is 


269.] 


ELLIPSOIDS   OF   INERTIA. 


147 


called  the  polar  reciprocal  of   the   former  surface  with  regard 
to  the  sphere. 

Let  Q  be  the  intersection  of  OP  with 
TT,  and  put  OP  =  p,  OQ  =  q\  then  it 
appears  from  the  figure  that 

pq  =  <?.  (16) 

269,  It  is  easy  to  see  that  the  polar 
reciprocal  of  the  momental  ellipsoid 
(n')  with  respect  to  the  sphere  of 
radius  e  is  the  ellipsoid 

oH o  i      o==  •*•  \*7/ 


Fig.  32. 


To  prove  this  it  is  only  necessary  to  show  that  the  relation  (16) 
is  fulfilled  for  p  as  radius  vector  of  (i  i'),  and  q  as  perpendicular 
to  the  tangent  plane  of  (17).  Now  this  tangent  plane  has  the 
equation 


hence  we  have  for  the  direction  cosines  «,  ft,  y,  and  for  the 
length  q,  of  the  perpendicular  to  the  tangent  plane 


These  relations  give    qla=(x/q^)qy 
whence 


For  the  radius  vector  p  of  (iif)  whose  direction  cosines  «,  y8,  7 
are  the  same  as  those  of  q  we  have  by  (iif)  : 


Hence  /o2^2=e4;  and  this  is  what  we  wished  to  prove. 


I48  KINETICS   OF   A   RIGID   BODY.  [270. 

270.  The  surface  (17)  has  variously  been  called  the  ellipsoid 
of  gyration,  the  ellipsoid  of  inertia,  the  reciprocal  ellipsoid.     We 
shall  adopt  the  last  name.     The  semi-axes  of  this  ellipsoid  are 
equal  to  the  principal  radii   of    inertia  at  the  point   O.     The 
directions    of    its  axes   coincide  with    those   of   the  momental 
ellipsoid  ;  but  the  greatest   axis  of  the  former   coincides  with 
the  least  of  the  latter,  and  vice  versa. 

By  comparing  the  equations  (12')  and  (18)  it  will  be  seen  that 
q  is  the  radius  of  inertia  of  the  line  («,  /?,  7)  on  which  it  lies. 
Thus,  while  the  radius  vector  OP  =  p  of  the  momental  ellipsoid  is 
inversely  proportional  to  the  radius  of  inertia,  i.e.  /o  =  e2/q,  the 
reciprocal  ellipsoid  gives  the  radius  of  inertia  <\for  a  line  1  as  the 
segment  cut  off  on  this  line  by  the  perpendicular  tangent  plane. 

271.  We   are   now  prepared   to  determine   the    moment    of 
inertia  for  any  line  in  space.     Let  us  construct  at  the  centroid 
G  of  the  given  mass  or  body  both  the  momental  ellipsoid  and 
its  polar  reciprocal.     The  former  is  usually  called  the  central 
ellipsoid  of  the  body ;  the  latter  we  may  call  the  fundamental 
ellipsoid  of  the  body.     As  soon  as  this  fundamental  ellipsoid 

^  +  ^  +  i=i 

q?   <?<?   3* 

is  known,  the  moment  of  inertia  of  the  body  for  any  line  what- 
ever can  readily  be  found.  For,  by  Art.  270,  the  radius  of 
inertia  q  for  any  line  /0  passing  through, the  centroid  is  equal 
to  the  segment  OQ  cut  off  on  the  line  /0  by  the  perpendicu- 
lar tangent  plane  of  the  fundamental  ellipsoid;  and  for  any 
line  /  not  passing  through  the  centroid  the  square  of  the 
radius  of  inertia  can  be  determined  by  first  finding  the  square 
of  the  radius  of  inertia  for  the  parallel  centroidal  line  /0  and 
then,  by  Art.  250,  adding  to  it  the  square  of  the  distance  d 
of  the  centroid  from  the  line  /. 

272.  In  the  problem  of  determining  the  ellipsoids  of  inertia 
for  a  given  body  at  any  point,  considerations  of  symmetry  are 


273-]  ELLIPSOIDS   OF   INERTIA. 

of  great  assistance,  similarly  as  in  the  problem  of  finding  the 
centroid  (compare  Part  II.,  Art.  47). 

Suppose  a  given  mass  to  have  a  plane  of  symmetry  ;  then 
taking  this  plane  as  the  jj/.s-plane,  and  a  perpendicular  to  it  as 
the  axis  of  x,  there  must  be,  for  every  particle  of  mass  m,  whose 
co-ordinates  are  x,  y,  2,  another  particle  of  equal  mass  m,  whose 
co-ordinates  are  —  x,  y,  z.  It  follows  that  the  two  products  of 
inertia  ^mzx  and  ^mxy  both  vanish,  whatever  the  position 
of  the  other  two  co-ordinate  planes.  Hence  any  perpendicular 
to  the  plane  of  symmetry  is  a  principal  axis  at  its  point  of  in- 
tersection with  this  plane. 

If  the  mass  have  two  planes  of  symmetry  at  right  angles  to 
each  other,  then  taking  one  as  ^-plane,  the  other  as  2-^-plane, 
and  hence  their  intersection  as  axis  of  x,  it  is  evident  that  all 
three  products  of  inertia  vanish, 


wherever  the  origin  be  taken  on  the  intersection  of  the  two 
planes.  Hence,  for  any  point  on  this  intersection,  the  principal 
axes  are  the  line  of  intersection  of  the  two  planes  of  symmetry, 
and  the  two  perpendiculars  to  it,  drawn  in  each  plane. 

If  there  be  three  planes  of  symmetry,  their  point  of  inter- 
section is  the  centroid,  and  their  lines  of  intersection  are  the 
principal  axes  at  the  centroid. 

273.   Exercises. 

Determine  the  principal  axes  and  radii  at  the  centroid,  the  central 
and  fundamental  ellipsoids,  and  show  how  to  find  the  moment  of  inertia 
for  any  line,  in  the  following  Exercises  (i),  (2),  (3). 

(  i  )  Rectangular  parallelepiped,  the  edges  being  2  a,  zb,  zc.  Find 
also  the  moments  of  inertia  for  the  edges  and  diagonals,  and  specialize 
for  the  cube. 

(2)  Ellipsoid  of  semi-axes  a,  b,  c.     Determine  also  the  radius  of 
inertia  for  a  parallel  /  to  the  shortest  axis  passing  through  the  extremity 
of  the  longest  axis. 

(3)  Right  circular  cone  of  height   h  and  radius  of  base  a.     Find 
first  the  principal  moments  at  the  vertex  ;  then  transfer  to  the  centroid. 


150  KINETICS   OF   A   RIGID   BODY.  [274. 

(4)  Determine  the  momental  ellipsoid  and  the  principal  axes  at  a 
vertex  of  a  cube  whose  edge  is  a. 

(5)  Determine  the  radius  of  inertia  of  a  thin  wire  bent  into  a  circle, 
for  a  line  through  the  centre  inclined  at  an  angle  a  to  the  plane  of  the 
circle. 

(6)  A  peg-top  is  composed  of  a  cone  of  height  H  and  radius  a,  and 
a  hemispherical  cap  of  the  same  radius.     The  point,  to  a  distance  h 
from  the  vertex  of  the  cone,  is  made  of  a  material  three  times  as  heavy 
as  the  rest.     Find  the  moment  of  inertia  for  the   axis   of  rotation ; 
specialize  for  h  =  a  —  \  H. 

(7)  Show  that  the  principal  axes  at  any  point  A,  situated  on  one  of 
the  principal  axes  of  a  body,  are  parallel  to  the  centroidal  principal  axes, 
and  find  their  moments  of  inertia. 

(8)  For  a  given  body  of  mass  M  find  the  points  at  which  the  mo- 
mental  ellipsoid  reduces  to  a  sphere. 

(9)  Determine  a  homogeneous  ellipsoid  having  the  same  mass  as  a 
given  body,  and  such  that  its  moment  of  inertia  for  every  line  shall  be 
the  same  as  that  of  the  given  body. 

3.    DISTRIBUTION    OF    PRINCIPAL   AXES    IN    SPACE. 

274.  It  has  been  shown  in  the  preceding  articles  how  the  principal 
axes  can  be  determined  at  any  particular  point.     The  distribution  of 
the  principal  axes  throughout  space  and  their  position  at  the  different 
points  is  brought  out  very  graphically  by  means  of  the  theory  of  con- 
focal  quadrics.     It  can  be  shown  that  the  directions  of  the  principal 
axes  at  any  point  are  those  of  the  principal  diameters  of  the  tangent 
cone  drawn  from  this  point  as  vertex  to  the  fundamental  ellipsoid ;  or, 
what  amounts  to  the  same  thing,  they  are  the  directions  of  the  normals 
of  the  three  quadric  surfaces  passing  through  the  point  and  confocal 
to  the  fundamental  ellipsoid. 

In  order  to  explain  and  prove  these  propositions  it  will  be  necessary 
to  give  a  short  sketch  of  the  theory  of  confocal  conies  and  quadrics. 

275.  Two  conic  sections  are  said  to  be  confocal  when  they  have  the 
same  foci.     The  directions  of  the  axes  of  all  conies  having  the  same 
two  points  S,  S'  as  foci  must  evidently  coincide,  and  the  equation  of 
such  conies  can  be  written  in  the  form 


:278.]  PRINCIPAL   AXES.  I$i 

where  X  is  an  arbitrary  parameter.  For,  whatever  value  may  be  assigned 
in  this  equation  to  A,  the  distance  of  the  centre  O  from  either  focus  will 
always  be  V«2  4-  X  —  (£2  -+-  X)  =  Vfl2  —  & ;  it  is  therefore  constant. 

276.  The  individual  curves  of  the  whole  system  of  confocal  conies 
represented  by  (19)  are  obtained  by  giving  to  X  any  particular  value 
between   —  oo  and   -f  oo  ;    thus  we  may  speak  of  the  conic  X  of  the 
system. 

For  X  =  o  we  have  the  so-called  fundamental  conic  x2/a2  +  y~/t>*  =  i  ; 
this  is  an  ellipse.  To  fix  the  ideas  let  us  assume  a~>b.  For  all  values 
of  X  >  —  £2,  i.e.  as  long  as  —  32  <  X  <  oo,  the  conies  (19)  are  ellipses, 
beginning  with  the  rectilinear  segment  SS'  (which  may  be  regarded  as 
a  degenerated  ellipse  X  =  —  ft  whose  minor  axis  is  o),  expanding  gradu- 
.ally,  passing  through  the  fundamental  ellipse  X  =  o,  and  finally  verging 
into  a  circle  of  infinite  radius  for  X  =  oo. 

It  is  thus  geometrically  evident  that  through  every  point  in  the  plane 
will  pass  one,  and  only  one,  of  these  ellipses. 

277.  Let  us  next  consider  what  the  equation  (19)  represents  when  X 
is  algebraically  less  than  —  bz.     The  values  of  X  that  are  <  —  a2  give 
imaginary  curves,  and  are  of  no  importance  for  our  purpose.     But  as 
long  as  —  a2  <  X  <  —  /52,  the  curves  are  hyperbolas.    The  curve  X  =  —  ft 
may  now  be  regarded  as  a  degenerated  hyperbola  collapsed  into  the 
two  rays  issuing  in  opposite  directions  from  S  and  S'  along  the  line  SS'. 
The  degenerated  ellipse  together  with  this  degenerated  hyperbola  thus 
represents  the  whole  axis  of  x. 

As  X  decreases,  the  hyperbola  expands,  and  finally,  for  X  =  —  a2,  verges 
into  the  axis  of  yt  which  may  be  regarded  as  another  degenerated 
hyperbola. 

The  system  of  confocal  hyperbolas  is  thus  seen  to  cover  likewise  the 
whole  plane  so  that  one,  and  only  one,  hyperbola  of  the  system  passes 
through  every  point  of  the  plane. 

278.  The  fact  that  every  point  of  the  plane  has  one  ellipse  and  one 
hyperbola  of  the  confocal  system  (19)  passing  through  it  allows  us  to 
regard   the  two  values   of  the  parameter  X  that  determine  these  two 
•curves  as  co-ordinates  of  the  point ;  they  are  called  elliptic  co-ordinates. 
If  x,  y   be    the    rectangular  Cartesian   co-ordinates   of  the   point,   its 
elliptic  co-ordinates  X1?  X2  are  found  as  the  roots  of  the  equation  (19) 
which   is  quadratic   in   X.      Conversely,   to   transform   from  elliptic  to 


152  KINETICS   OF   A   RIGID   BODY.  [279. 

Cartesian  co-ordinates,  that  is,  to  express  x  and  y  in  terms  of  Xj  and  X2r 
we  have  only  to  solve  for  x  and  y  the  two  equations 


279.  The  two  confocal  conies  that  pass  through  the  same  point  P 
intersect  at  right  angles.     For  the  tangent  to  the  ellipse  at  P  bisects  the 
exterior  angle  at  P  in  the  triangle  SPS',  while  the  tangent  to  the  hyper- 
bola bisects  the  interior  angle  at  the  same  point ;  in  other  words,  the 
tangent  to  one  curve  is  normal  to  the  other,  and  vice  versa.    The  elliptic 
system  of  co-ordinates  is,  therefore,  an  orthogonal  system ;  the  infinitesi- 
mal elements  d\l  •  d\2  into  which  the  two  series  of  confocal  conies  (19) 
divide  the  plane  are  rectangular,  though  curvilinear. 

280.  These   considerations  are  easily  extended  to  space   of  three 
dimensions. 

An  ellipsoid 

-i+^+Z^  =  i,  where  a  >*><:, 
a       o      c 

has  six  real  foci  in  its  principal  planes ;  two,  .Si,  .Si',  in  the  dry-plane,  on> 
the  axis  of  x,  at  a  distance  (2Si  =  V02  —  ft2  from  the  centre  O ;  two, 
S2,  S2',  in  the  jyz-plane,  on  the  axis  of  y,  at  the  distance  OS2  =  V^2  —  t3' 
from  the  centre ;  and  two,  S3,  S3)  in  the  &#-plane,  on  the  axis  of  x,  at 
the  distance  OS3  —  vV  —  c2  from  the  centre.  It  should  be  noticed 
that,  since  b  >  c,  we  have  OS3  >  0.Si ;  i.e.  Slf  SJ  lie  between  S3)  S3'  on 
the  axis  of  x. 

The  same  holds  for  hyperboloids. 

281.  Two  quadric  surfaces  are  said  to  be  confocal  when  their  princi- 
pal sections  are  confocal  conies.     Now  this  will  be  the  case  for  two- 
quadric  surfaces  whose  semi-axes  are  alf  b^  c^  and  az,  £2,  cz,  if  the 
directions  of  their  axes  coincide  and  if 


Writing  these  conditions  in  the  form 

,,2          -2 Z2          Z2 -2          -2     cav   \ 

a1    —  a\    =  &2    —  Pi    —  ^2    —  C\  )  SaY   —  A> 

we  find   «22  =  a}  +  X,   £22  =  <V  +  X,   r22  =  c?  +  X.     Hence  the  equation 

x2 


284.]  PRINCIPAL   AXES.  153 

where  X  is  a  variable  parameter,  represents  a  system  of  confocal  quad- 
ric  surfaces. 

282.  As  long  as  X  is  algebraically  greater  than  —  <r2,  the  equation 
(20)  represents  ellipsoids.     For  A  =  —  c2  the  surface  collapses  into  the 
interior  area  of  the  ellipse  in  the  jry-plane  whose  vertices  are  the  foci 
S2,  -5V  and  SB,  Ss-     For  as  A  approaches  the  limit  —  t2,  the  three  semi- 
axes  of  (20)  approach  the  limits  V#2  —  ^2,  V^2  —  t2,  o,   respectively. 
This  limiting  ellipse  is  called  the  focal  ellipse.     Its  foci  are  the  points 
Sb.Si1,  since  a2  -  c~  -  (b2  —  cz)  =  a2  -  P. 

When  X  is  algebraically  <  —  c2,  but  >  —  a2,  the  equation  (20)  repre- 
sents hyperboloids ;  for  values  of  X  <  —  a2  it  is  not  satisfied  by  any  real 
points.  As  long  as  —  b2<  A<  —  c2,  the  surfaces  are  hyperboloids  of  one 
sheet.  The  limiting  surface  X  =  —  c2  now  represents  the  exterior  area 
of  the  focal  ellipse  in  the  ary- plane.  The  limiting  hyperboloid  of  one 
sheet  for  X  =  —  b2  is  the  area  in  the  sac-plane  bounded  by  the  hyperbola 
whose  vertices  are  .Si,  £/,  and  whose  foci  are  S3,  S8'.  This  is  called  the 
focal  hyperbola. 

Finally,  when  —  a2  <  A  <  —  b2,  the  surfaces  are  hyperboloids  of  two 
sheets,  the  limiting  hyperboloid  X  =  —  a2  collapsing  into  the  jz-plane. 

283.  It  appears  from  these  geometrical  considerations,  that  there 
are  passing  through  every  point  of  space  three  surfaces  confocal  to  the 
fundamental  ellipsoid  x2/a2  +y*/b2  +  z2/c2  =  i  and  to  each  other,  viz. : 
an  ellipsoid,  a  hyperboloid  of  one  sheet,  and  a  hyperboloid  of  two  sheets. 
This  can  also  be  shown  analytically,  as  there  is  no  difficulty  in  proving 
that  the  equation  (20)  has  three  real  roots,  say  \lt  A2,  A3,  for  every  set 
of  real  values  of  x,  y,  z)  and  that  these  roots  are  confined  between  such 
limits  as  to  give  the  three  surfaces  just  mentioned. 

The  quantities  A1;  A2,  A3  can  therefore  be  taken  as  co-ordinates  of  the 
point  (x,  y,  z)  ;  and  these  elliptic  co-ordinates  of  the  point  are,  geomet- 
rically, the  parameters  of  the  three  quadric  surfaces  passing  through 
the  point  and  confocal  to  the  fundamental  ellipsoid ;  while,  analytically, 
they  are  the  three  roots  of  the  cubic  (20).  To  express  x,  y,  z  in  terms 
of  the  elliptic  co-ordinates,  it  is  only  necessary  to  solve  for  x,  y,  z  the 
three  equations  obtained  by  substituting  in  (20)  successively  Xlf  A2,  A$ 
for  A. 

284.  The  geometrical  meaning  of  the  parameter  A  will  appear  by 
considering  two  parallel  tangent  planes  TTO  and  TTA  (on  the  same  side  of 


KINETICS    OF   A   RIGID    BODY.  [285. 

the  origin),  the  former  (TTO)  tangent  to  the  fundamental  ellipsoid 
^y02_|_yy£2_|_2y£2_.  I?  the  latter  (TTA)  tangent  to  any  confocal  surface 
A  or  oi/(a*  +  A)  +  //(£2  +  X)  +  z2/(^  +  A)  =  i.  The  perpendiculars 
^0,  ^x,  let  fall  from  the  origin  O  on  these  tangent  planes  TTO,  TTA,  are  given 
by  the  relations  (the  proof  being  the  same  as  in  Art.  269). 

qf  =  <*<*  +  Pp  +  cV,  (21) 

q*=  (a2  +  AK  +  O*2  +  A)/?2  +  (^  +  A)y2,  (22) 

where  a,  /?,  y  are  the  direction-cosines  of  the  common  normal  of  the 
planes  7r0,  TTA.  Subtracting  (21)  from  (22),  we  find,  since  a2-f/82-|-y2=  i, 

?A2-?o2  =  A;  (23) 

i.e.  the  parameter  A  of  any  one  of  the  confocal  surfaces  (20)  is  equal  to 
the  difference  of  the  squares  of  the  perpendiculars  let  fall  from  the  common 
centre  on  any  tangent  plane  to  the  surface  A,  and  on  the  parallel  tangent 
plane  to  the  fundamental  ellipsoid  \  =  o. 

285.  Let  us  now  apply  these  results  to  the  question  of  the  distri- 
bution of  the  principal  axes  throughout  space. 

We  take  the  centroid  G  of  the  given  body  as  origin,  and  select  as 
fundamental  ellipsoid  of  our  confocal  system  the  polar  reciprocal  of  the 
central  ellipsoid,  i.e.  the  ellipsoid  (17)  formed  for  the  centroid,  for 
which  the  name  "  fundamental  ellipsoid  of  the  body"  was  introduced  in 
Art.  271.  Its  equation  is 


if  q-b  q%,  <?3  are  the  principal  radii  of  inertia  of  the  body. 

The  radius  of  inertia  qQ  for  any  centroidal  line  4  can  be  constructed 
(Art.  2  70)  by  laying  a  tangent  plane  to  this  ellipsoid  perpendicular  to 
the  line  /0  ;  if  this  line  meets  the  tangent  plane  in  Q0  (Fig.  33),  then 
^0=  GQo-  Analytically,  if  a,  ft,  y  be  the  direction-cosines  of  /0,  q§  is 
given  by  formula  (21)  or  (12'). 

286.  To  find  the  radius  of  inertia  q  for  a  line  /,  parallel  to  /0,  and 
passing  through  any  point  P,  we  lay  through  P  a  plane  TTA,  perpendicular 
to  /,  and  a  parallel  plane  TTO,  tangent  to  the  fundamental  ellipsoid;  let 
<2A,  (?o  be  the  intersections  of  these  planes  with  the  centroidal  line  /0. 
Then,  putting  £<2o=?o,  GQ^=qx,  GP=r,  PQ^  =  d,  we  have,  by 
Art.  250, 


289.] 


PRINCIPAL   AXES. 


155 


The  figure  gives  the  relation  dz  =  r1  —  q£,  which,  in  combination  with 
(23),  reduces  the  expression  for  the  radius  of  inertia  for  the  line  /  to 
the  simple  form  : 

?2=^-A.  (24) 

287.  The  value  of  r2  —  X,  and  hence  the  value  of  q,  remains  the  same 
for  the  perpendiculars  to  all  planes  through  P,  tangent  to  the   same 
quadric  surface  X  :  these  per- 

pendiculars   form,    therefore, 

an  equimomental  cone  at  P. 

By  varying  A.  we  thus  obtain 

all  the   equimomental    cones 

at  P.     The  principal  diame- 

ters of  all  these  cones  coin- 

cide in  direction,  since  they 

coincide  with  the    directions 

of  the  principal  axes  of  the 

momental  ellipsoid  at  P  (see 

Art.  265);  but  they  also  coin- 

cide with  the  principal  diam- 

eters of  the  cones  enveloped  Fig.  33. 

t>y  the  tangent  planes  TTA.     It 

thus  appears  that  the  principal  axes  at  the  point  P  coincide  in  direction 

with  the  principal  diameters  of  the  tangent  cone  from  P  as  vertex  to 

the  fundamental  ellipsoid  x2/q?  +f/q?  +  #/$$  =  i. 

288.  Instead  of  the  fundamental  ellipsoid,  we  might  have  used  any 
quadric  surface  A.  confocal  to  it.     In  particular,  we  may  select  the  con- 
focal  surfaces  A*,  A.2,  X3  that  pass  through  P.     For  each  of  these  the  cone 
of  the  tangent  planes  collapses  into  a  plane,  viz.  the  tangent  plane  to 
the  surface  at  P,  while  the  cone  of  the  perpendiculars  reduces  to  a  single 
line,  viz.  the  normal  to  the  surface  at  P.     Thus  we  find  that  the  prin- 
cipal axes  at  any  point  P  coincide  in  direction  with  the  normals  to  the 
three  quadric  surfaces,  confocal  to  the  fundamental  ellipsoid  and  passing 
through  P. 

For  the  magnitudes  of  the  principal  radii  qx,  qy,  qz  at  P,  we  evidently 
have 


289.   Exercise. 

(i)    The  principal  radii  q±,  q^  q%  of  a  body  being  given,  find  the 
equation  of  the  momental  ellipsoid  at  any  point  P,  referred  to  axes 


!0  KINETICS    OF   A   RIGID   BODY.  [290 

through  this  point  P  parallel  to  the  principal  axes  of  the  body ;  deter- 
mine the  directions  of  the  principal  axes  at  P,  and  show  that  these 
directions  coincide  with  the  normals  of  the  three  surfaces  passing 
through  P  and  confocal  to  the  fundamental  ellipsoid  of  the  body. 

290.   A  brief  account  of  the  theory  of  moments  of  inertia  will  be  found 
in  B.  WILLIAMSON,  Integral  Calculus,  6th  ed.,  London,  Longmans,  1891, 
pp.    291-312.     The  subject  is  discussed  very   fully  in   E.  J.  ROUTH,    j 
Dynamics  of  a  system  of  rigid  bodies,  Part  I.,  5th  ed.,  London,  Mac-    j 
millan,  1891,  pp.  1-49;  and  in  B.  PRICE,  Analytical  mechanics,  Vol.  II.,. 
2d  ed.,  Oxford,  Clarendon  Press,  1889,  Chapter  IV.     The  student  will   1 
also  consult  with  advantage  W.  SCHELL,  Theorie  der  Bewegung  und  der  1 
Krdfte,  Vol.  I.,   2d    ed.,  Leipzig,  Teubner,    1879,   PP-   100-143,  an<i   1 
A.    CAYLEY,   Report  on  the  progress  of  the  solution  of  certain  special 
problems  of  dynamics,  in  the  Report  of  the  32d  meeting  of  the  British 
Association,  for  1862,  London,  Murray,  1863,  pp.  223-229. 


293-]  BODY   WITH   FIXED   AXIS.  157 

III.    Rigid  Body  ^vith  a  Fixed  Axis. 

291.  A  rigid  body  with  a  fixed  axis  has  but  one  degree  of 
freedom.     Its  motion  is  fully  determined  by  the  motion  of  any 
one  of  its  points  (not  situated  on  the  axis),  and  any  such  point 
must  move  in  a  circle  about  the  axis.     Any  particular  position 
of  the  body  is,  therefore,  determined   by  a  single  variable,  or 
co-ordinate,  such  as  the  angle  of  rotation.     Just  as  the  equi- 
librium of  such  a  body  depends  on  a  single  condition  (see  Part 
IL,  Art.  227),  so  its  motion  is  given  by  a  single  equation. 

292.  The  equation   of  motion  can  be  derived  directly  from 
the  proposition  of  angular  momentum  (Art.  224).     Let  r  be 
the  distance  of  any  particle  m  of  the  body  from  the  fixed  axis, 
to  the  angular  velocity  at  the  time  t ;  then  mwr  is  the  momentum 
of  the  particle,  and  mwr1  its  moment,  or  the  angular  momentum 
of  the  particle,  about  the  axis.     At  any  given  instant  t,  CD  has 
the  same  value  for  all  particles.     Hence,  the  angular  momentum 
of  the   body  is   w^mr*  =  o>7,  where  I=^mr^  is  the  moment  of 
inertia  of  the  body  for  the  fixed  axis. 

Now,  by  Art.  224,  the  rate  at  which  the  angular  momentum 
of  the  body  about  the  axis  changes  with  the  time  is  equal  to 
the  sum  of  the  moments  of  all  the  external  forces  about  the 
.same  axis.  Denoting  this  resulting  moment  by  H,  and  con- 
sidering that  the  moment  of  inertia  for  the  fixed  axis  is  inde- 
pendent of  the  time,  we  have  the  equation  of  motion 

dv_H.  (} 

dt~  T 

i.e.  the  angular  acceleration  about  the  fixed  axis  is  equal  to  the 
moment  of  all  the  external  forces  about  this  axis,  divided  by  the 
moment  of  inertia  of  the  body  for  the  same  axis. 

293.  The  same  result   can    of  course  be  obtained  from  any 
one  of   the  equations  (6)  or   (7),  Art.  224.     Thus,  taking   the 
fixed  line  as  axis  of  2,  the  third  of  the  equation  (7),  viz. 

—  ^ 
at 


I58  KINETICS  OF   A   RIGID    BODY.  [294. 

must  be  used.     Now,  for  rotation  of  angular  velocity  co  about 
the  axis  of  #,  we  have  ^=  —  (*>y,  y  =  wx.     Hence 

^m  (xy  —yx)  —  w^m  (x*  -fj/2)  =  w'Snir*  =  &>/. 
The  equation  assumes,  therefore,  the  form  (i). 

294.  The  reactions  of  the  fixed  axis  do  not  enter  into  the 
composition  of  the  resulting  moment  H.  As  they  intersect  the 
axis,  their  moments  about  this  axis  are  zero. 

The  student  should  notice  the  close  analogy  between  equa- 
tion ^  and  the  equation  for  the  rectilinear  motion  of  a  particle, 

dv_F 
j.  —  > 
at  m 

where  v  is  the  velocity  and  F  the  resultant  of  all  the  forces  act- 
ing on  the  particle. 

The  expression  for  the  kinetic  energy  of  a  body  rotating  about 
a  fixed  axis  is 

T=S%mv*=2%mrfr*  =  %fco*,  (2) 

and  has  also  a  form  similar  to  that  for  the  kinetic  energy  of  a 
particle  m  moving  with  velocity  v  in  a  straight  line,  viz. 


295.  Let  us  denote  the  angle  of  rotation  by  0,  so  that 
<*>  =  d6/dt,  d<ti/dt=d2d/dt\  If  the  resulting  moment  be  con- 
stant or  a  given  function  of  6,  say  //=/(#),  the  equation  of 
motion 


can  be  integrated  once,  and  gives 


where  &>0  is  the  angular  velocity  corresponding  to  the  angle  00. 

This  is  the  equation  of  kinetic  energy.  It  might  have  been 
derived  directly,  according  to  Art.  234,  by  expressing  that  the 
increase  of  the  kinetic  energy  equals  the  work  of  the  forces. 


297-] 


BODY   WITH    FIXED   AXIS. 


159 


The  kinetic  energy  is  given  by  (2).  The  work  of  a  force  F  in 
a  plane  perpendicular  to  the  axis,  at  the  distance/  from  the  axis, 
is  F-pdO  for  an  infinitesimal  rotation  of  angle  dO  ;  hence,  the  sum 
of  the  elementary  works  of  all  the  forces  = 


296.  While  thus  the  motion  of  a  rigid  body  about  a  fixed  axis 
is  given  by  a  single  equation,  the  other  equations  of  motion  of  a 
rigid  body  are  required  to  determine  the  reactions  of  the  fixed  axis 
(comp.  Part  II.,  Art.  227). 

The  axis  will  be  fixed  if  any  two  of  its  points  A,  B  are 
fixed.  The  reaction  of  the  fixed  point  A  can  be  resolved  into 
three  components  Ax,  Ay,  Ag,  that  of  B 
into  Bx,  By,  Bx.  By  introducing  these  re- 
actions the  body  becomes  free  ;  and  the 
system  composed  of  these  reactions,  of  the 
external  forces,  and  of  the  reversed  effec- 
tive forces  must  be  in  equilibrium.  We 
take  again  the  axis  of  rotation  as  axis  of  z 
(Fig.  34)  so  that  the  ^-co-ordinates  of  the 
particles  are  constant,  and  hence  z=o, 
i?  =  o;  and  we  put  OA=a,  OB=b.  Then 
the  six  equations  of  motion  are  (see  Art. 
223  (4)  and  Art.  224  (6))  : 


Fig.  34. 


^m  (xy  —yx)  =  *Z(x  Y—yX). 

297.  It  remains  to  introduce  into  these  equations  the  values 
for  x,  y.  As  the  motion  is  a  pure  rotation,  we  have  (see  Part  I., 
Art.  245)  x= —<oy,  y  =  a*x\  hence,  x=  —  u>y  —  aPx, y=zwx—aPy. 
Summing  over  the  whole  body,  we  find 

^mx  =  —  &)  %my  —  <tF%mx  =  —  Mwy  —  May^x, 
'=     Mwx—Mw*y, 


KINETICS   OF   A   RIGID    BODY.  [298. 

-where  x,y  are  the  co-ordinates  of  the  centroid  ;  and 


z  =  —  Ew  +  Deo2, 

x  =  —  a&myz  —  aPSmzx  =  —  Diet  —  Ea>2, 
y  —yx)  =  a&mx2  —  aP^mxy  +  a&my*  +  w^mxy  =  Ceo, 

where  C=2m(x2+y2),    D  =  *Zmyz,    E=^mzx  are  the  notations 
introduced  in  Art.  255. 

With  these  values  the  equations  of  motion  assume  the  form  : 

-  Myu  =  ?.X+AX  +  Bx, 


-zY)-  aAy  - 
-xZ]  +  aAx 


298.    The  last  equation  is  identical  with  equation  (i). 

The  components  of  the  reactions  along  the  axis  of  rotation 
•occur  only  in  the  third  equation,  and  can  therefore  not  be  found 
separately.  The  longitudinal  pressure  on  the  axis  is 

=  -A.-S.=*Z. 

The  remaining  four  equations  are  sufficient  to  determine  At 
Ay,  Bx,  By. 

The  total  stress  to  which  the  axis  is  subject,  instead  ol:  being 
resolved  into  two  forces,  at  A  and  B,  can  be  reduced  for  the 
origin  O  to  a  force  and  a  couple  (see  Fig.  34).  The  equations 
(4)  give  for  the  components  of  the  force 

-AX-BX  =  2X+ 

(5) 


This  force  consists  of  the  resultant  of  the  external  forces, 
R  = 


and  two  forces  in  the  ^/-plane  which  form  the  reversed  effective 
force  of  the  centroid  ;  for  Mxw2  and  MyuP  give  as  resultant  the 


299-]  BODY   WITH   FIXED   AXIS.  l6l 


centrifugal  force  J/a)2V^2-f-J/2  =  J/a)V,  directed  from  the  axis 
towards  the  projection  of  the  centroid  on  the  ;rj/-plane,  while 
Mywy—Mxw  form  the  tangential  resultant  Ma*r,  perpendicular  to 
the  plane  through  axis  and  centroid. 

The  couple  has  a  component  in  the  ^-plane,  and  one  in  the 
^-plane,  viz.  : 

aAy  +  bBy  = 


(6) 
-  aAn  -  bBx  =  ^(zX-xZ}  +  Eco*  +  Da, 


while  the  component  in  the  ^/-plane  is  zero.  The  resultant 
•couple  lies,  therefore,  in  a  plane  passing  through  the  axis  of 
rotation. 

299.  In  the  particular  case  when  no  forces  X,  Y,  Z  are  acting 
on  the  body,  the  last  of  the  equations  (4),  or  equation  (i),  shows 
that  the  angular  velocity  e»  remains  constant.  The  stress  on  the 
axis  of  rotation  will,  however,  exist  ;  and  the  axis  will  in  general 
tend  to  change  both  its  direction,  owing  to  the  couple  (6),  and 
its  position,  owing  to  the  force  (5). 

If  the  axis  be  not  fixed  as  a  whole,  but  only  one  of  its  points, 
the  origin,  be  fixed,  the  force  (5)  is  taken  up  by  the  fixed  point, 
while  the  couple  (6)  will  change  the  direction  of  the  axis.  Now 
this  couple  vanishes  if,  in  addition  to  the  absence  of  external 
forces,  the  conditions 

D=^myz=o,    E=^mzx=o  (7) 

are  fulfilled.  In  this  case  the  body  would  continue  to  rotate 
about  the  axis  of  z  even  if  this  axis  were  not  fixed,  provided  that 
the  origin  is  a  fixed  point.  A  line  having  this  property  is  called 
a  permanent  axis  of  rotation. 

As  the  meaning  of  the  conditions  (7)  is  that  the  axis  of  z  is  a 
principal  axis  of  inertia  at  the  origin  (see  Art.  264),  we  have  the 
proposition  that  if  a  rigid  body  with  a  fixed  point,  not  acted  upon 

any  forces,  begin  to  rotate  about  one  of  the  principal  axes  at 
this  point,  it  will  continue  to  rotate  uniformly  about  the  same 
•axis.  In  other  words,  the  principal  axes  at  any  point  are 

PART   III  —  II 


KINETICS   OF  A   RIGID   BODY. 


[300. 


always,  and  are  the  only,  permanent  axes  of  rotation.     This  can 
be  regarded  as  the  dynamical  definition  of  principal  axes. 

300.  It  appears  from  the  equations  (5)  that  the  position  of 
the  axis  of  rotation  will  remain  the  same  if,  in  addition  to  the 
absence  of  external  forces,  the  conditions 

^•=0,  j/  =  o  (8) 

be  fulfilled  ;  for  in  this  case  the  components  of  the  force  (5)  al 
vanish.  If,  moreover,,  the  axis  of  rotation  be  a  principal  axis 
the  rotation  will  continue  to  take  place  about  the  same  line  even 
when  the  body  has  no  fixed  point. 

The  conditions  (8)  mean  that  the  centroid  lies  on  the  axis  ol 
z\  and  it  is  known  (Art.  264)  that  a  centroidal  principal  axis  is 
a  principal  axis  at  every  one  of  its  points.  The  axis  of  z  must 
therefore  be  a  principal  axis  of  the  body,  i.e.  a  principal  axis  at 
the  centroid.  We  have,  therefore,  •  the  proposition  :  If  a  fret 
rigid  body,  not  acted  upon  by  any  forces,  begin  to  rotate  about  one 
of  its  centroidal  principal  axes,  it  will  contimte  to  rotate  uniformly 
about  the  same  line. 

301.  A  rigid  body  with  a  fixed  horizontal  axis  is  called  a 
compound  pendulum   if  the   only   external   force  acting  is  the 
weight  of  the  body. 

The  plane  through  axis  and  centroid  will  make,  with  the 
vertical  plane  (downwards)  through  the  axis,  an  angle  0,  which 
we  may  take  as  angle  of  rotation,  so  that 
a)  =  d6/dt  (Fig.  35).  The  weights  of  the  par- 
ticles, being  all  parallel  and  proportional  tc 
their  masses,  have  a  single  resultant  Mg  pass 
ing  through  the  centroid  G.  Hence,  if  h  be 
the  perpendicular  distance  OG  of  the  centroic 
from  the  axis,  the  moment  of  the  externa 
forces  is  H=  —Mgh  sin  6  ;  and  if  the  radius  oi 
inertia  of  the  body  for  the  centroidal  axis 
parallel  to  the  axis  of  rotation  be  q,  the  moment 
of  inertia  for  the  latter  axis  is  I= 


Ma 


Fig.  35. 


303.]  BODY   WITH    FIXED   AXIS.  ^3 

With   these   values   the   equation   of   motion  (i)  assumes   the 
simple  form 

h      . 

(9) 


As  shown  in  Art.  175,  the  equation  of  the  simple  pendulum 
of  length  /  is 

d*0         g  •    Q 

|.  ^=-5sm* 

The  two  equations  differ  only  in  the  constant  factor  of  sin  0, 
and  it  appears  that  the  motion  of  a  compound  pendulum  is  the 
same  as  that  of  a  simple  pendulum  whose  length  is 


302.  The  problem  of  the  compound  pendulum  has  thus  been 
reduced  to  that  of  the  simple  pendulum.     The  length  /  is  called 
the   length  of  the   equivalent    simple  pendulum.     The    foot    O 
(Fig.  35)  of  the  perpendicular  let  fall  from  the  centroid  on  the 
axis   is  called  the  centre  of  suspension.     If  on  the  line  OG  a 
length   OC=-l  be  laid  off,   the  point   C  is  called  the  centre  of 
oscillation.     It  appears,  from  (10),  that  G  lies  between  O  and  C. 

The  relation  (10)  can  be  written  in  the  form 

h(l-K)=q\  or  OG-  6Y7=const. 

As  this  relation  is  not  altered  by  interchanging  O  and  C,  it 
follows  that  the  centres  of  oscillation  and  suspension  are  inter- 
changeable ;  i.e.  the  period  of  a  compound  pendulum  remains 
the  same  if  it  be  made  to  swing  about  a  parallel  axis  through 
the  centre  of  oscillation. 

303.  Exercises, 

(1)  A  pendulum,  formed  of  a  cylindrical  rod  of  radius  a  and  length 
Z,  swings  about  a  diameter  of  one  of  the  bases.     Find  the  time  of  a 
small  oscillation. 

(2)  A  cube,  whose  edge  is  a,  swings  as  a  pendulum  about  an  edge. 
Find  the  length  of  the  equivalent  simple  pendulum. 


164  KINETICS    OF   A   RIGID    BODY.  [304. 

(3)  A  circular  disc  of  radius  r  revolves  uniformly  about  its  axis, 
making  100  revolutions  per  minute.     What  is  its  kinetic  energy? 

(4)  A  fly-wheel  of  radius  r,  in  which  a  mass,  equal  to  that  of  the  disc 
in  Ex.  (3),  is  distributed  uniformly  along  the  rim,  has  the  same  angular 
velocity  as  the  disc.     Neglecting  the  mass  of  the  nave  and  spokes, 
determine  its  kinetic  energy,  and  compare  it  with  that  of  the  disc. 

(5)  A  fly-wheel  of  12  ft.  diameter,  whose  rim  weighs  12  tons,  makes 
50  revolutions  per  minute.     Find  its  kinetic  energy  in  foot-pounds. 

(6)  A  fly-wheel  of  radius  r  and  mass  m  is  making  ^V  revolutions  per 
minute  when  the  steam  is  shut  off.     If  the  radius  of  the  shaft  be  r\  and 
the  coefficient  of  friction  /u,,  find  after  how  many  revolutions  the  wheel 
will  come  to  rest  owing  to  the  axle  friction. 

(7)  A  fly-wheel  of  10  ft.  diameter,  weighing  5  tons,  is  making  40 
revolutions  when  thrown  out  of  gear.     In  what  time  does  it  come  to 
rest  if  the  diameter  of  the  axle  is  6  in.  and  the  coefficient  of  friction 
fj.  =  0.05  ? 

(8)  A  uniform  straight  rod  of  length  /is  hinged  at  one  end  so  as  to 
turn   freely  in  a  vertical  plane.     If  it  be  dropped  from  a  horizontal 
position,  with  what  angular  velocity  does  it  pass  through  the  vertical 
position?     (Equate  the  kinetic  energy  to  the  work  of  gravity.) 

304.  Impulses.  Suppose  a  rigid  body  with  a  fixed  axis  is 
acted  upon,  when  at  rest,  by  a  single  impulse  F,  in  a  plane 
perpendicular  to  the  axis  and  at  the  distance  /  from  the  axis. 
It  is  required  to  determine  the  initial  motion  of  the  body  just 
after  impact. 

As  the  impulsive  reactions  of  the  fixed  axis  have  no  moment 
about  this  axis,  the  initial  angular  momentum  of  the  body  about 
the  fixed  axis  must  be  equal  to  the  moment  of  the  impulse  F 
about  the  same  axis ;  i.e.  to  Fp.  If  co  is  the  initial  angular 
velocity,  the  momentum  of  a  particle  m  at  the  distance  r  from 
the  axis  is  mwr\  hence  the  angular  momentum  of  the  body 
=  2ma>r*=co2mr2=ci)f,  where  /  is  the  moment  of  inertia  of  the 
body  for  the  fixed  axis.  Hence  we  have 

<*=¥-•  (ii) 


3o6.]  BODY  WITH   FIXED   AXIS.  165 

305.  Let  the  impulse  F  be  produced  by  the  inelastic  impact 
of  a  particle  of  mass  m  moving  with  a  velocity  u.  It  would  not 
be  correct  to  put  F=mu  in  (11);  as  the  particle  after  impact 
continues  to  move  with  the  body  with  a  certain  velocity  v,  it 
does  not  actually  give  up  to  the  body  its  whole  momentum, 
but  only  the  amount  F—m(u  —  v),  provided  that  u  and  v  have 
the  same  direction.  With  this  assumption,  which  evidently 
means  that  the  particle  meets  the  body  atTsome  point  of  the 
plane  passing  through  the  axis  and  perpendicular  to  u,  the 
velocity  after  impact  is  v  =  a>p.  With  the  value  (11)  of  o>  this 
gives 


whence  F=muf/(f+mp2),  and  finally,  by  (n), 


(12) 


As  mp^  is  the  moment  of  inertia  of  the  particle  for  the  fixed 
axis,  this  formula  shows  that  we  may  substitute  in  (11)  the 
whole  momentum  mu  for  F  if  we  increase  the  moment  of 
inertia  of  the  body  by  that  of  the  particle ;  in  other  words, 
that  the  particle  may  be  regarded  as  giving  up  its  whole 
momentum  if  it  be  taken  into  account  that  after  impact  it 
forms  part  of  the  body. 

306.  It  is  easy  to  see  how  the  considerations  of  the  last  two 
articles  can  be  generalized.  When  any  number  of  impulses 
act  in  various  directions  on  a  rigid  body  with  a  fixed  axis,  the 
initial  angular  velocity  will  be  determined  by 

«=f  >  (13) 

where  H  is  the  sum  of  the  moments  of  all  the  impulses  about 
the  fixed  axis. 


i66 


KINETICS   OF   A   RIGID   BODY. 


[307. 


307.  To  determine  the  impulsive  stress  produced  on  the  axis 
by  a  single  impulse  F,  let  us  write  out  the  general  equations 
of  the  impulsive  motion. 

Take  the  fixed  axis  as  the  axis  of  z  and  the  ^jtr-plane  through  the 
centroid  G  (Fig.  36),  and  let  ^,  o,  o  be  the  co-ordinates  of  G,  and 


z 
B* 

*B 


z\  tnose  °f  tne  point  o; 


°v 


1/7 


X 


Fig.  36. 


application  P  of  the  impulse 
The  components  of  F  may  be 
denoted  by  X,    Y,  Z ;  those  o 
the  reactions  of  the  axis  by  Ax 
Ay,  Az,  Bx,  By,  Bg,  similarly  as 
in  Art.  296. 

~ As  the  initial  motion  after 
impact  is  a  rotation  about  the 
axis  of  z,  we  have  x=  —  c 
y  =  <ax,  2=0,  so  that  the  mo 
mentum  of  a  particle  of  mass 
m  has  the  components  —may,  max,  o.  Reducing  these  mo 
menta  to  the  origin  (9,  we  find  a  resultant  momentum  whose 
components  are  —  w^my=o,  <£&mx=Mwx,  o;  and  a  resulting 
couple  whose  vector  has  the  components  —  co^mzx=—L 
—  (t)^myz=—D(D,  a)^m(x^-\-y2')  =  C(i))  where  C,  D,  E  have  the 
same  meaning  as  in  Art.  297. 

The  six  equations  of  motion  just  after  the  impulse  are  there 
fore,  if  the  body  was  originally  at  rest : 


(14) 


Ca=x1Y-y1X. 

308.   The  last  of  these  equations  is  nothing  but  the  equation 
(i  i).     The  components  A,,  B,  along  the  axis  cannot  be  deter- 


Y+Ay+Byt 


—  z^  Y—  aAy  —  bBy, 


309.]  BODY   WITH    FIXED   AXIS. 

mined  separately;  the  other  components  of  the  reactions  can 
be  found  from  the  first,  second,  fourth,  and  fifth  equations. 

The  impulsive  stress  to  which  the  axis  is  subjected  by  the 
impulse,  or  the  so-called  percussion  of  the  axis,  instead  of  being 
represented  by  two  impulses  A,  B  as  above,  can  also  be  regarded 
as  composed  of  an  impulse  whose  components  are 


-AX-BX  =  X,  -Ay-By  =  Y-  , 

and  an  impulsive  couple  whose  vector  has  the  components 

o. 


The  last  component  being  zero,  the  resulting  couple  lies  in  a 
plane  passing  through  the  axis  of  2. 

If  there  were  any  number  of  impulses  acting  on  the  body 
simultaneously,  the  effect  on  the  axis  could  be  determined  in 
the  same  way,  except  that  the  quantities  X,  Y,  Z,  y^Z—z^Yt 
z^X—x^Z,  must  be  replaced  by  the  corresponding  sums. 

309.  It  follows  from  the  preceding  article  that  the  conditions 
under  -which  a  single  impulse  acting  on  a  rigid  body  with  a 
fixed  axis  will  produce  no  stress  on  the  axis  are 


Z=o,    -zlMx+E=o,   D=o.      (15) 

If  these  conditions  are  fulfilled,  the  resulting  motion  will  be  the 
same  even  when  the  axis  is  free. 

The  first  and  third  equations  show  that  the  impulse  must  be 
perpendicular  to  the  plane  passing  through  axis  and  centroid. 
The  meaning  of  the  fourth  and  fifth  conditions  becomes  appar- 
ent if  the  .rj/-plane  be  taken  so  as  to  pass  through  the  point  of 
application  P  of  the  impulse.  The  new  origin  O'  is  the  foot  of 
the  perpendicular  let  fall  from  P  on  the  fixed  axis.  To  trans- 
form the  conditions  (15)  to  the  new  system  it  is  only  necessary 
to  substitute  z+zl  for  z\  the  first  three  conditions  are  not 
affected,  and  the  last  two  become 

—  z-^Mx  +  3  mzx  +  z^mx  =  o,    *%myz  +  z^my  =  o, 


KINETICS   OF   A   RIGID   BODY.  [310. 

or,  since  *$mx=Mx,  2w/  =  o, 

E'  =  o,   Z?'=o, 

where  Ef,  D'  are  the  products  of  inertia  at  <9'. 

It  thus  appears  that  the  axis  of  z  must  be  a  principal  axis  at 
the  foot  of  the  perpendicular  let  fall  on  this  axis  from  the  point 
of  application  of  the  impulse. 

310.  It  should  be  noticed  that  a  line  taken  at  random  in  a 
body  is  not  necessarily  a  principal  axis  at  any  one  of  its  points. 
But  if  a  line  is  a  principal  axis  at  a  point  O',  then  it  is  always 
possible  to  determine  an  impulse  that  will  produce  no  stress  on 
this  line  so  that  the  body  will  begin  to  rotate  about  it  as  axis 
even  though  it  be  not  fixed.  As  shown  in  the  last  article,  the 
impulse  must  be  =Mxayy  and  must  be  directed  at  right  angles  to 
the  plane  through  axis  and  centroid.  The  point  where  it  meets 
this  plane  is  called  the  centre  of  percussion.  Its  distance  x±  from 
the  axis  is  found  from  the  equation  of  motion,  viz.  the  last  of 
the  equations  (14)  which,  owing  to  the  conditions  (15),  reduces  to 


If  q1  be  the  radius  of  inertia  of  the  body  for  a  parallel  centroidal 
axis,  we  have  C=M(q'*+x*)  ;  hence 

x^x+VJ-.  (16) 

Hence,  if  a  given  line  /  be  principal  axis  for  one  of  its  points  O1  ', 
there  exists  a  centre  of  percussion  ;  it  lies  on  the  intersection 
of  the  plane  (/,  G)  with  the  plane  through  Of  perpendicular  to  /, 
at  the  distance  xv  given  by  (16),  from  the  line  /.  An  impulse 
Mxo)  through  the  centre  of  percussion  at  right  angles  to  the 
plane  through  axis  and  centroid,  while  producing  no  percussion 
on  the  axis,  sets  the  body  rotating  with  angular  velocity  a>  if  it 
was  originally  at  rest  ;  on  the  other  hand,  if  the  body  wz 
originally  in  rotation  about  the  axis,  such  an  impulse  can  brinj 
the  body  to  rest  without  affecting  the  axis. 


3I4-]  BODY  WITH   FIXED   POINT. 


IV.    Rigid  Body  with  a  Fixed  Point. 

311.  A  rigid  body  with  a  fixed  point  has  three  degrees  of 
freedom.     Any  one  of    its  points,   with  the  exception  of  the 
fixed  point  O,  is  constrained  to  the  surface  of  a  sphere  and  has 
therefore  two  degrees  of  freedom  ;  and  the  body  itself  can  turn 
about  the  line  joining  this  point  to  O.     The  motion  consists,  at 
any  instant,  of  an  infinitesimal  rotation  about  an  axis  passing 
through  O  (see  Part  L,  Arts.  32-35).    Both  the  angular  velocity 
and  the  direction  of  the  instantaneous  axis  vary  in  the  course 
of  time. 

312.  We  begin  with  the  study  of  the  instantaneous  motion  of 
the  body,  which  may  be  regarded  as  due  to  the  action  of  a  sys- 
tem of  impulses  on  the  body  at  rest.     This  will  lead  to  the  solu- 
tion of  the  converse  problem,  viz.  to  determine  the  initial  motion 
produced  by  a  given  system  of  impulses. 

I.     INITIAL    MOTION    DUE    TO    IMPULSES. 

313.  The  body  rotates  at  the  time  /with  angular  velocity  o> 
about  the  instantaneous  axis  /  which  passes  through  the  fixed 
point  O.    It  is  required  to  determine  a  system  of  impulses  that 
would  produce  this  motion  if  acting  on  the  body  at  rest. 

For  O  as  origin,  let  R  be  the  resultant  and  H  the  result- 
ing couple  of  these  impulses.  If  the  impulsive  reaction  A  of 
the  fixed  point  O  be  combined  with  them,  the  body  can  be 
regarded  as  free,  and  its  instantaneous  motion  is  determined  by 
the  equations  (19)  and  (20),  Art.  238.  It  is  only  necessary,  in 
the  equations  (19),  to  add  to  the  components  RXJ  Ry,  Rz  of  R 
those  of  A,  while  the  right-hand  members  of  (20)  are  not 
affected  by  A,  since  its  moment  is  zero  for  every  axis  through  O. 

314.  It  remains  to  form  the  sums  in  the  left-hand  members 
of    (19)    and    (20)  for  our  case  ;  i.e.  to  reduce   the  system  of 
momenta  mx,  my,  mz  of  the  particles  to  its  resultant  and  result- 
ant couple  for  a  fixed  rectangular  system  of  axes  through  O. 


KINETICS   OF   A   RIGID    BODY.  [315. 

The  resultant  momentum  has  evidently  the  components 


where  Jr,  y,  ~z  are  the  components  of  the  velocity  of  the  cen- 
troid  at  the  time  /,  and  Mis  the  mass  of  the  body.  Hence  the 
equations  (19)  become 


(i) 

These  equations  serve  to  determine  the  impulsive  pressure 
—  A  =  —  ^A^+Af+A?  on  the  fixed  point  O  in  magnitude  and 
direction. 

315.  To  form  the  moment  ^m(yz—zy)  of  the  momenta  of 
the  particles,  i.e.  the  angular  momentum  of  the  body,  about  the 
axis  of  xy  we  resolve  the  angular  velocity  co  into  its  components 
ox,,  ft)y,  cot  along  the  axes  and  observe  that  the  components  of  the 
linear  velocity  of  any  point  (x,  y,  z]  arising  from  the  rotation 
.are  (Part  I.,  Art.  293)  : 


Substituting  these  values,  we  find 

2  m  (y'z  —  zy)  =  w^my2  —  (o^mxy  —  a^mzx  + 
or  with  the  notation  of  Art.  255, 

2m  (y'z  —  zy)=Awx  —  Fw>y  —  Ewt. 

Forming  in  the  same  way  the  angular  momenta  about  the  axes 
of  y  and  z>  we  find  the  equations  (20)  in  the  form 

A  cox  —  Fcoy  —  E(DZ  =  Hn 

-  Fa>x  +  B<*y  -  D<»z  =  Hy,  (2) 

—  £cox  —  Dwy  -f  Ccoz  =  Hz. 

316.  It  appears,  then,  that  the  rotation  of  angular  velocity 
co  about  the  axis  1  can  be  regarded  as  due  to  an  impulsive  couple 
H  whose  components  are  given  by  (2).  Conversely,  the  effect  of 


3I7-]  BODY   WITH   FIXED   POINT.  1?i 

a  couple  H  on  a  rigid  body  at  rest,  with  a  fixed  point,  is  to 
impart  to  the  body  a  rotation  CD  whose  magnitude  and  axis  can 
be  found  by  determining  cox)  coy)  coz  from  (2). 

Any  system  of  impulses  acting  on  the  body  can  be  reduced,  for 
the  fixed  point  (9  as  origin,  to  a  resultant  R  and  a  couple  H  ;  the 
effect  of  the  couple  has  just  been  stated  ;  that  of  R  consists  merely 
in  producing  a  pressure  on  the  fixed  point.  To  find  this  pressure, 
determine  e>=  Vo)x2-f  u£-\-w?  from  (2);  the  velocity  of  the  cen- 
troid  can  then  be  found  and  its  components  substituted  in  (i), 
Art.  314. 

317.  The  axis  /  of  the  rotation  produced  by  a  given  couple 
H  is  not,  in  general,  perpendicular  to  the  plane  of  the  couple. 
Imagine  the  angular  velocity  co  to  be  represented  by  its  rotor, 
i.e.  by  a  length  co  laid  off  from  O  on  the  axis  /,  and  the  couple  H 
by  its  vector,  i.e.  by  a  length  //'laid  off  from  O  on  the  perpen- 
dicular to  the  plane  of  the  couple.  The  relation  between  the 
rotor  co  and  the  vector  H  producing  it  will  best  appear  if  we  take 
the  axis  of  rotation  /  as  axis  of  z.  We  then  have  x—  —  coy, 
y  =  cox,  z=o,  and  the  momenta  —mcoy,  mcox,  o  of  the  particles 
reduce  to  a  resultant  and  couple  at  O  as  follows.  The  resultant 
momentum  has  the  components  : 

—  county  =  —  My  co,     co^mx  =  Mxco,     O  ; 

it  is  equal  to  Mco  V^2  4-  y2  =  Mar,  where  r  is  the  distance  of  the 
centroid  from  the  axis  /,  and  is  perpendicular  to  the  plane 
through  axis  and  centroid.  The  couple  has  the  components 

—  oftmzx=  —  Eco,      —  u&myz  =  —  Deo,     co^m  (x*  +y2)  =  Ceo. 
The  equations  (19)  and  (20)  of  Art.  238  reduce  therefore  to 


Mxco  =Ry+Ay,         o  =Rz+Ag,          (3) 
-Eco    =HX,  -Dco  =  Hy,  Cco  =  Hx.  (4) 


These  equations  can  also  be  derived  directly  from  the  equations 
(i)  and  (2)  above,  since  in  the  present  case  we  have  x=—  coy, 


KINETICS   OF   A   RIGID   BODY. 


318.   The  equations  (4)   show  that,    in   general,    the   couple 
H  has  three  components  (see  Fig.   37);   Hx  and  Hy  can   be 

combined     into     a     partial     re- 
sultant   7/L,,  = 


H 


in    the 

and  the  total  resultant 
+  D*  +  £z  makes  with 
the  axis  /  an  angle  $  such  that 

As  C 


H 


Fig.  37. 


is  always  positive,  this  angle  is 
always  acute ;  it  vanishes  only 
if  D=o  and  E=o,  i.e.  if  the 
instantaneous  axis  /  is  a  principal 
axis  at  O. 

This  result  that  H  and  o>  coincide  only  along  a  principal  axis 
is  very  important.  It  shows  that  the  vector  H  of  the  couple  that 
produces  a  rotation  w  has  the  direction  of  tJie  axis  of  rotation  1 
only,  and  always,  if  this  axis  1  is  a  principal  axis  at  the  fixed 
point  O  ;  in  this  case  we  have  H  =  Io>,  where  I  is  the  moment  of 
inertia  for  \. 

Conversely,  a  couple  H  acting  on  a  rigid  body  with  a  fixed 
point  O  produces  rotation  about  an  instantaneous  axis  /,  which 
is,  in  general,  inclined  to  the  vector  of  the  couple  at  an  acute 
angle  <£.  This  angle  reduces  to  zero,  i.e.  the  instantaneous  axis 
/  coincides  in  direction  with  the  vector  of  the  couple,  only, 
and  always,  when  the  plane  of  the  couple  is  perpendicular  to  a 
principal  axis  at  O. 

319.  Let  us  now  take  the  principal  axes  at  O  as  axes  of 
co-ordinates.  Let  wj,  a>2,  &>3  be  the  components  of  o>  along 
these  axes;  //i»  •%»  ^3  those  of  H ' ;  and  let  7P  72,  73  be  the 


principal  moments, 
Then  we  must  have 


qz  the  principal  radii  of  inertia  at  O. 


These  relations  follow  also  from  (2),  since  A=fv  B  —  IV  C=I3, 
Z>  =  o,  E=o,  F=o;  they  determine  the  relation  between  Zfand 
o)  in  the  general  case. 


320.] 


BODY   WITH   FIXED   POINT. 


173 


320.  The  relation  between  the  vectors  H  and  o>  is  very 
clearly  brought  out  by  making  use  of  the  ellipsoids  of  inertia 
at  the  point  O. 

The  reciprocal  ellipsoid  at  O  has  the  equation  (see  Arts. 
269,  270) 


Let  P  (Fig.  38)   be  the  point  where  it  is  met  by  the  vector 
H\  x,  y,  z  the  co-ordinates,  p  the  radius  vector  of  P\    hence 


Fig.  38. 


•*/P*  y/P>  2/P 
H^  =  Hy/p,  H 


whence 


Fig.  39. 

direction-cosines   of  H,  so  that  H^  —  l 
=  Hz/p.     The  equations  (5)  give,  therefore, 

-£...£.        =tL.iL    m=fL.JL 

Mp  '  q\*    ^     Mp  '  q£'       3     Mp  *  qf 


(6) 


where  q  is  the  perpendicular  let  fall  from   O  on  the  tangent 
plane  at  P  (see  Art.  269).     The  direction-cosines  of  CD, 


are  the  same  as  those  of  this  perpendicular  (ib.). 

It  thus  appears  that  the  plane  through  O  at  right  angles  to  the 
instantaneous  axis  1  is  conjugate  to  the  direction  of  the  vector  H 
with  respect  to  the  reciprocal  ellipsoid  at  O. 


KINETICS   OF   A   RIGID   BODY.  [321. 

321.   Again,  the  equation  of  the  momental  ellipsoid  at  O  is 
(see  Art.  266) 


the  semi-axes  being  a  =  e*/gl,  b 

Let  the  instantaneous  axis  /  meet  this  ellipsoid  at  a  point  Pf 
(Fig.  39)  whose  co-ordinates  and  radius  vector  are  x',  y\  s',  p', 
so  that  x'/p',  y'/p',  z'/p'  are  the  direction-cosines  of  /  and 
to^wx'/p',  Q>2  =  a>y//3f,  (03  =  (oz'/p'.  Substituting  these  values 
and  introducing  the  semi-axes  a,  b,  ct  we  find  form  (5) 


„  .  . 

whence  ~ir+¥+s  =  -7-  '  ?• 

where  ^'  is  the  perpendicular  let  fall  from  O  on  the  tangent 
plane  at  P\     The  direction-cosines  of  H, 


agree  with  those  of  q\ 

It  follows  that  the  plane  of  the  couple  H  is  conjugate  to  the 
direction  of  the  instantaneous  axis  1  with  respect  to  the  momental 
ellipsoid  at  O. 

322.  The  kinetic  energy  of  a  rigid  body  with  a  fixed  point  O 
has  the  expression 


where  ^mr2  is  the  moment   of   inertia  for  the  instantaneous 
axis  /. 

Now,  by  Art.  270,  the  radius  of  inertia  for  the  line  /  is  equal 
to  the  distance  of  O  from  the  perpendicular  tangent  plane  to 
the  reciprocal  ellipsoid,  i.e.  to  q  (Fig.  38).  Hence 

(8) 


322.]  BODY   WITH    FIXED   POINT.  ^5 

As,   according  to  the  fundamental  property  of  the  momental 
ellipsoid  (see  (16),  Art.  266),  we  have  q  —  e2/p\  this  becomes 

/      .'         (9) 

On  the  other  hand,  by  Art.  258,  if  «,  &  7  be  the  direction- 
cosines  of  /,  z>.  of  the  rotor  a),  we  have 


hence 

T=$(A<»*  +  Bto*+  Ceo*-  2  Da>ycoz-2  Ecozcox-2  Fcoxcoy).    (10) 

Differentiating  with  respect  to  cox,  coy,  coz,  and  comparing  with 
the  equations  (2),  we  find 


-=     A»m-F»,-  £».=#„ 

da)x 

H,,  (ii) 


y 


If  these  relations  be  multiplied  by  a)x,  coy,  wz  and  added,  they 
give 

2  Tco^co.+co^H^+tf^  +  ff^.        (12) 


Substituting  in  the  last  expression  aco  =  cox,  /3co  =  coy,  yco  =  coz,  or 
\H=HX,  pH=Hy,  vH=Hz,  where  X,  jj,,  v  are  the  direction- 
cosines  of  the  vector  H,  we  find 

We  have,  therefore,  for  the  projection  of  H  on  the  instan- 
taneous axis  /, 

2  T 
H  cos  (f>  =  f?xct.  H-  ffyj3  +  H.fl  =  —  =  Mcp'to  j 

CO 

for  the  projection  of  o>  on  the  direction  of  the  vector  H, 

2T  9o)2 


z— —  - 

H  H 

and  finally,  T=  J  //o>  cos  </>.  (14) 


KINETICS   OF   A   RIGID   BODY.  [323. 

323.  If  the  principal  axes  at  O  be  taken  as  axes  of  co-ordi- 
nates, we  have  to  write  <ov  <o2,  o>3  for  <ax,  <oy,  taz;  Hlt  /72,  7/3  for 
.HM  Hy,  Hn\  7lf  72,  73  for  At  B,  C,  while  Z>  =  o,  £=o,  F=o. 
Thus  the  relations  (5)  give  //1  =  71eo1,  7/2  =  72o>2,  7/3  =  73a>3,  whence 


SL  +  SL-  (l6) 

and  7^=  ^  (T^wj  4-  72o)2  +  73o>3 )  (17) 

(18) 

For  the  angle  <£  between  7f  and  o>,  we  have 

COS  ffl  =  -*-"* — ^ ~.      —  =  — ^ — •*• rr 

7/0)  7/o> 

2.    CONTINUOUS    MOTION    UNDER   ANY    FORCES. 

324.  We  now  proceed  to  consider  the  motion  of  a  rigid  body 
with  a  fixed  point  when  acted  upon  by  any  forces. 

For  the  fixed  point  O  as  origin,  the  external  forces  reduce  to 
a  resultant  R  and  a  couple  H.  While  the  force  R  is  taken  up 
by  the  fixed  point,  the  effect  of  the  couple  consists  in  changing 
the  angular  velocity  o>  about  the  instantaneous  axis  /,  which 
exists  at  the  time  /,  to  the  angular  velocity  o>4-</o)  about 
another  instantaneous  axis  /',  which  determines  the  motion  of 
the  body  at  the  time  t  +  dt.  The  point  O  being  fixed,  both 
axes,  /  and  /,  pass  through  it ;  and  by  Part  I.,  Art.  303,  the 
acceleration  of  any  point  (x,  y,  z)  of  the  body  has  the  following 
components  parallel  to  rectangular  axes  fixed  in  the  body  and 
moving  with  it  : 


x = a>x  (mxx  4- 

y  =  &y(coxx+c0yy  +  co^)—(o2y  +  Q)gx—Q)xz,  (l) 

'z  =  c0g((t)xx  4-  Q)y  y  -h  o)^)  —  o)2^  4-  cox y  —  ufx. 


326.]  BODY   WITH   FIXED   POINT. 

Multiplying  these  expressions  by  the  mass  m  of  the  particle 
situated  at  the  point  (x,  y,  z),  we  have  the  components  of  the 
effective  force  of  this  particle. 

325.  To  form  the  equations  of  motion  (4),  Art.  223,  and  (6), 
Art.  224,  for  our  case,  we  must  reduce  the  system  of  the 
effective  forces  to  its  resultant  and  resulting  couple  ;  or,  what 
amounts  to  the  same  thing,  we  must  form  the  sums  occurring 
in  the  left-hand  members  of  these  equations. 

The  summation  of  the  components  of  the  effective  forces 
throughout  the  body  gives,  as  usual, 

^mx  —  MX,    ^  my  —  My,    ^mz  =  Mz, 

where  x,  y,  ~z  are  the  components  of  the  acceleration  of  the 
centroid.  The  resultant  is  therefore  equal  to  the  effective  force 
of  the  centroid,  the  whole  mass  M  of  the  body  being  regarded 
as  concentrated  at  this  point. 

To  make  the  body  free,  the  reaction  A  of  the  fixed  point 
should  be  introduced.  Denoting  its  components  by  Ax,  Ay,  Agt 
those  of  the  resultant  R  of  the  external  forces  by  Rx,  Ry,  Rg,  the 
equations  (4),  Art.  223,  assume  the  form 


(2) 

The  left-hand  members  evidently  vanish  if  the  origin  be  the 
centroid.  The  equations  (2)  can  serve  to  determine  the  press- 
ure —  A  on  the  fixed  point  in  magnitude  and  direction. 

326.  To  form  the  moment  ^m(y'z-zy)  of  the  effective  forces 
about  the  axis  of  x,  we  have  to  multiply  the  second  of  the 
expressions  (i)  by  z,  and  subtract  the  product  from  the  third 
multiplied  by  y  ;  then  multiply  the  difference  by  m,  and  sum 
throughout  the  body. 

Performing  this  operation  first  on  the  last  two  terms  which 
were  shown  in  Part  L,  Art.  302,  to  be  due  to  the  angular 
acceleration,  we  find 


PART   III  —  12 


KINETICS   OF   A   RIGID   BODY.  [327. 

with  the  notation  of  Art.  255.  As  the  axes  are  fixed  in  the 
body  (Art.  324),  the  moments  and  products  of  inertia  are  con- 
stant ;  and  it  appears  from  the  equations  (2),  Art.  315,  that  this 
expression  is  the  derivative  with  respect  to  the  time  of  the 
component  Hx  of  the  impulsive  couple  H  that  produces  the 
rotation  &>  at  the  time  /. 

Next  operating  in  the  same  way  on  the  remaining  terms  of 
the  component  accelerations  (i),  viz.  those  arising  from  the 
centripetal  acceleration,  we  find 


w^myz)  — 
+  co^mz2)  +  o^myz 
F(Dx  +  Ccoy  +  Da>,}  -  coy(Ecox  +  D<o,  +  Ba>.) 

—  E(i)x  —  D(dy  +  Cd)g)  —  Q)g(  —  F(0X  +  B(Dy  — 


by  (2),  Art.  315. 

The  moments  of  the  effective  forces  about  the  other  two  axes 
can  now  be  obtained  by  cyclical  permutation  of  the  subscripts 
x,  y,  z.  Thus  we  find  that  the  equations  (6),  Art.  224,  assume 
the  form 


(3) 


The  reaction  A  of  the  fixed  point  does  not  enter  into  these 
equations  ;  as  it  intersects  every  one  of  the  axes,  its  moments 
about  these  axes  are  zero. 


327.  Geometrically  the  equations  (3)  mean  that  the  vector  #of  the 
resultant  couple  of  the  external  forces  has  two  components  one  of  which 
resolves  itself  along  the  axes  into  dHx/dt,  dHJdt,  dHJdt,  while  the 


328-] 


BODY   WITH    FIXED   POINT. 


179 


wrffy  — 


other    has    the    components    ^yHz—  (az£fy,   uzHx—  u>xHz, 

Each    of   these  components    can    be    inter- 

preted geometrically,  if  we  imagine  the  vector 

H  of  the  impulsive  couple  drawn  from  O  as 

origin,  so  that  the  co-ordinates  of  its  extremity 

are  Hx,  Hy,  Hz  (Fig.  40).     The  time-deriva- 

tives of  these  co-ordinates  are  the  velocities 

of  the  extremity  of  the  vector  H  with  respect 

to  the  axes  of  co-ordinates  which,  it  will  be 

remembered,  are  fixed  in  the  body.      Hence 

that  component  of  H  which  is  due  to   the 

angular  acceleration  is  the  relative  velocity  of 

the  extremity  of  H  with  respect  to  the  body. 

The  other  component,  which  is  due  to  the  centripetal  acceleration, 
evidently  represents  the  linear  velocity,  arising  from  the  angular  velocity 
<o,  of  the  point  of  the  body  that  coincides  at  the  time  /  with  the  same 
extremity  of  the  vector  H. 

It  follows  that  the  vector  H  represents  in  magnitude  and  direction 
the  absolute  velocity  of  the  extremity  of  the  vector  H  '•  in  other  words, 
H  is  geometrically  equal  to  the  geometrical  increment  of  H  divided  by 
the  element  of  time.  This  was  to  be  expected,  and  might  indeed  be 
taken  as  starting-point  for  deriving  the  equations  (3). 


Fig.  49. 


328.  Let  us  now  select  as  axes  of  co-ordinates  the  principal 
axes  at  O.  According  to  our  usual  notation,  we  have  then  to 
exchange  the  subscripts  x,  y,  z  for  i,  2,  3.  Moreover,  as  shown 
in  Art.  319,  H^I^  ff2  =  I2co2,  /73  =  73a>3,  where  7lf  72,  73  are 
the  principal  moments  of  inertia  at  O.  Thus  the  equations  (3) 
reduce  to  the  following  : 


a)l  +  (78  - 


(4) 


73d>3  +  (72  -  7^  o)1c»2  =  7/g. 

These  are  Euler's  equations  of  motion.     Their  solution  gives 
o)2,  o>3  as  functions  of  the  time  t. 


!8o  KINETICS   OF   A   RIGID    BODY.  [329. 

It  may  be  noted  that  the  equations  (4)  are  often  written  with  the 
following  notation  : 


at 

C)rp=M,  (4') 


where  A,  B,  C  are  the  principal  moments  of  inertia  ;  /,  g,  r  the  com- 
ponents of  the  angular  velocity  o>  along  the  principal  axes  ;  Z,  M,  N 
the  components  of  the  resulting  couple  H  of  the  external  forces  along 
the  same  axes. 

329.  Owing  to  the  importance  of  the  equations  (4)  it  may  be  well 
to  indicate  another  way  of  deriving  them. 

The  rotation  of  angular  velocity  to  about  the  instantaneous  axis  /  dur- 
ing the  first  element  of  time  can  be  regarded  as  due  to  an  impulsive 
couple  H  (Art.  316).  Even  if  there  were  no  external  forces  acting,  the 
body  would  not  in  general  continue  to  turn  with  the  same  velocity  about 
the  same  axis.  For  if  this  were  the  case,  any  particle  m  of  the  body, 
at  the  distance  r  from  the  axis  /would  be  moving  uniformly  in  a  circle 
of  radius  r,  with  a  velocity  <ar,  and  such  uniform  circular  motion 
requires  for  its  maintenance  the  action  of  a  centripetal  force. 

Let  us  therefore  introduce  at  every  particle  m  two  equal  and  opposite 
forces  (Fig.  41),  the  centripetal  force  ma?r  directed  towards  the  axis  /, 
and  the  centrifugal  force  —  m<*?r\  the  intro- 
duction of  these  forces  does  not  change  the 
state  of  motion  of  the  body. 

330.  If  the  system  of  centripetal  forces 
nn*?r  alone  were  introduced  and  no  other 
forces  were  acting,  the  body  would  continue 
to  turn  with  the  same  angular  velocity  <u  about 
the  same  axis  /.  The  effect  of  the  system  of 
centrifugal  forces  —ma?r  represents  therefore 
the  change  that  would  take  place  in  the 
Fig.  41.  motion  if  no  external  forces  were  acting. 

Let  us  reduce  these   centrifugal   forces  to 

their  resultant  and  resulting  couple,  the  fixed  point  O  being  taken  as 
origin  and  the  axis  /  as  axis  of  z.  In  doing  this  we  can  make  use  of 


332.]  BODY   WITH   FIXED   POINT.  181 

the  reduction  of  momenta  in  Art.  317.  For,  evidently,  the  vector 
representing  the  centrifugal  force  —  muPr  can  be  obtained  by  multiply- 
ing the  momentum  mwr  of  the  particle  m  by  cu  and  turning  it  through 
an  angle  of  90°  in  a  sense  opposite  to  that  of  the  rotation  <o.  The 
reduction  to  O  gives  therefore  a  resultant  force  M^r,  in  the  ary-plane, 
directed  toward  the  projection  of  the  centroid  on  this  plane.  The 
resulting  couple  has  its  2-component  equal  to  zero  since  all  the  centrifu- 
gal forces  intersect  the  axis  of  z;  the  vector  of  the  resulting  couple 
lies  therefore  in  the  xy-  plane,  has  the  magnitude  &Hxyy  and  is  perpen- 
dicular to  the  Hxy  in  Fig.  37. 

331.  The  resultant  vanishes  only  if  ?  =  o,  i.e.  if  the  centroid  lies  on 
the  axis  /;  the  couple  vanishes  if  uHxy  =  <o2  VZ>2  -f-  £2  =  o,  i.e.  if  the 
axis  /  is  a  principal  axis  at   O.     It  follows  that  the  centrifugal  forces 
reduce  to  zero  only  if  the  axis  of  rotation  is  a  principal  centroidal  axis  ; 
in  this  case  the  direction  of  the  axis  remains  unchanged. 

By  Art.  318  (see  Fig.  37)  we  have  Hxy  =  7/sin  <£  ;  hence  the  result- 
ing couple  of  the  centrifugal  forces  =  w/f  sin  <£,  that  is,  its  magnitude  is 
represented  by  the  area  of  the  parallelogram  formed  by  the  vectors  H 
and  a)  ;  the  vector  of  this  couple  as  shown  above  is  perpendicular  to 
this  area.  Projecting  this  parallelogram  on  any  three  rectangular  co-ordi- 
nate planes,  with  O  as  origin,  we  find,  since  <ax,  <ay,  <az  are  the  co-ordi- 
nates of  the  extremity  of  the  rotor  <o,  HM  Hy,  Hy  those  of  the  extremity 
of  the  vector  H  drawn  from  O  : 

<ae  Hy  —  wy  HMt    wx  Hg  —  <oz  Hxt    o>y  Hx  —  <ajfr 
This  agrees  with  the  results  found  in  Arts.  326,  327. 

332.  If  the  principal  axes  at  O  be  taken  as  axes  of  co-ordinates,  the 
components  of  the  resultant  couple  of  the  centrifugal  forces  become 


or,  since  ^i  =  /!(!>!,    H^—I^^    .//3=/3eo3, 

(72  —  73)  (020)3,       (73  —  /i)  ftfcCDi,       (/i 

As  the  planes  of  these  couples  are  perpendicular  to  the  principal  axes  at 
O,  they  produce  during  the  element  of  time  infinitesimal  rotations  about 
these  axes,  whose  angles  are,  by  Art.  318  : 


/I 


182 


KINETICS   OF   A   RIGID    BODY. 


[333- 


These  are  the  only  increments  of  talt  w2,  o>3  if  there  are  no  forces 
acting  on  the  body  ;  hence,  in  this  case  we  must  have 


-       =  (72  — 


/3  -        =  (/!  —  72)  W^. 


If,  however,  there  are  external  forces  acting  on  the  body,  whose  result- 
ing couple  for  O  is  £f,  with  the  components  fflt  H2,  7/3  along  the  prin- 
cipal axes  at  O,  these  couples  produce  infinitesimal  rotations 


and  the  equations  of  motion  are  therefore 
7xwi  =  (72  — 
^2  =  (73  — 

=  (7i  —  72)  <Diu2  4-  7/3. 
These  are  Euler's  equations  (4). 

333.    Euler's  equations  determine  the  angular  velocities  of 
the  body  about  the  principal  axes  which  move  with  the  body. 

The  position  of  these  moving 
axes  with  respect  to  a  system 
of  fixed  rectangular  axes 
through  the  fixed  point  O 
can  be  expressed  by  means 
of  three  angles. 

Let  X,  Y,  Z  (Fig.  42)  be 
the  intersections  of  the  fixed 
axes,  with  a  sphere  of  radius 
one,  described  about  O  as 
centre;  X\  Y\  Z'  those  of 
the  moving  principal  axes ; 
N  the  intersection  with  the 

same  sphere  of  the  so-called  line  of  nodes,  i.e.  the  line  in  which 
the  planes  JT<9Fand  X'OY'  intersect.     Then  the  angles 


usually  called  Euler's  angles,  may  serve  to  determine  the  relation 
between  the  two  systems  of  axes. 


335-]  BODY  WITH   FIXED   POINT. 

334.  The  sense  in  which  these  angles  are  counted  is  best  remembered 
by  imagining  the  two  trihedral  angles  X YZ  and  X'  Y'Z'  originally  coin- 
cident.    Now  turn  the  system  X'Y'Z'  about  the  axis  OZ  in  the  posi- 
tive sense  (counter-clockwise)  until  the  axis   OX'  coincides  with  the 
assumed  positive  sense  of  the  nodal  line   ON,  i.e.  the  final  intersection 
•of  the  planes  XOY  and  X'OY' ;  the  amount  of  this  rotation  gives  the 
angle  {j/.     Next  turn  the  trihedral  X fY'Z'  about  this  line  of  nodes  in 
the  positive  sense  until  the  plane  X'OY'  falls  into  its  final  position; 
this  gives  the  angle  6,  as  the  angle  between  the   planes  XOY  and 
.X'OY'  at  N,  or  as  the  angle  ZOZ'  between  their  normals.     Finally  a 
rotation  of  X'Y'Z1  about  the  axis  OZ',  which  has  reached  its  final  posi- 
tion, in  the  positive  sense  until   OX1  comes   into   its  final  position, 
•determines  the  angle  <f>. 

335.  The  angular  velocity,  represented  by  its  rotor  CD,  whose 
•components  along  OX',  OY',  OZ'  are  co^  o>2,  &>3,  can  be  resolved 
.along  ON,  OZ1,  OZ  into  three  components  which  are  evidently 
0,  c/>,  ^,  respectively.     The  sum  of  the  projections  of  these  three 
components  on  the  line  OX'  should  give  o>l ;  hence 

cw1  =  0  cos  (j>  -f  <j)  cos  JTT  4-  ^  cos  ZX'. 
.Similarly         o>2  =  0  cos  ($  +  JTT)  -j-  </>  cos  |-TT  +  ^r  cos  Z  Y', 

0)3  =  0  COS  |-  7T  H-  (j)  COS  O  +  ^r  COS  0. 

The  spherical  triangle  ZNX'  gives  (by  the  fundamental  formula 
•of  spherical  trigonometry,  cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cosy) 
•cos  ZX '  =  sin  0  cos  (J  TT  —  0)  =  sin  <f>  sin  0 ;  and  the  triangle  ZNY' 
gives  cos^F^sin^-hl-TrJcosd-Tr  —  0)  =  cos  <p>  sin  0.  Hence, 
iinally 


a)l  =    cos     +  ir  sn     sn    , 

o>2  =  —  6  sin  <£  +  ^  cos  <£  sin  0,  ($) 

G>3  =  0  +  ^  COS0. 

.Solving  these  equations  for  0,  <^,  njr,  we  find 
0  =  ft)1  cos  <£  —  w2  sin  <^>, 

</>= —ojj  sin  <^>  cot  0  —  o)2  cos  <^>  cot  0+®3»  (6) 

^  =  o)1  sin  (^  esc  0-fft>2  cos  </>  esc  0. 


1 84 


KINETICS   OF   A   RIGID   BODY. 


[336. 


336.  The  relation  between  two  rectangular  systems  of  axes 
with  the  same  origin  can  also  be  expressed  by  means  of  the 
9  cosines  of  the  angles  between  the  axes. 

Let  O  be  the  common  origin,  x>  y,  z  the  co-ordinates  of  any 
point  with  respect  to  the  fixed  system,  x\  y\  2'  its  co-ordinates 
in  the  moving  system ;  then  we  have,  evidently, 


(7) 


where  the  coefficients  of  x\  y\  z'  are  the  cosines  of  the  angles 
between  the  axes,  which  can  best  be  remembered  in  the  form 


X1 

y' 

z' 

X 

*l 

a* 

as 

y 

b\ 

^2 

&3 

z 

c\ 

CcL 

C3 

Thus,  9  angles  are  used  in  order  to  fix  the  position  of  the 
moving  axes  with  respect  to  the  fixed  axes,  instead  of  Euler's 
3  angles.  But  their  9  cosines  (8)  are  connected  by  6  in- 
dependent relations,  which  can  be  written  in  either  one  of  the 
equivalent  forms  : 


or 


I, 


°- 


The  meaning  of  these  equations  is  easily  perceived  from  the 
meaning  of  the  angles  involved.  Thus,  the  first  of  the  equations, 
(9)  expresses  the  fact  that  alf  blt  c^  are  the  direction-cosines 


338.]  BODY   WITH   FIXED   POINT.  185 

of  a  line,  viz.  the  axis  Ox1  ;  the  last  of  the  equations  (10) 
expresses  the  perpendicularity  of  the  axes  'Ox  and  Oy  ;  and 
similarly  for  the  others. 

337.  The  relations  between  the  9  angles,  whose  cosines  are 
given  in  (8),  and  Euler's  3  angles  6,  <£,  -\/r  are  readily  found  from 
Fig.  42,  by  applying  the  formula  cos  c=  cos  a  cos  b  +  sin  a  sin  b  cosy 
successively  to  the  triangles 

XNX',  XNY',  XNZ', 

YNX',  YNY',  YNZ\ 

ZNX',  ZNY1,  ZNZ1. 

In  this  way  the  following  relations  are  found  : 
^  =  cos  i|r  cos  </>  —  sin  ty  sin  <f>  cos  6, 
&!  =  sin  i/r  cos  $  +  cos  ^r  sin  <£  cos  0, 
cl  =  sin  <f)  sin  6, 
a%=  —  cos  'v/r  sin  <^>  —  sin  -^  cos  (/>  cos  ^,       a3  =  sin  A/r  sin  6, 


.  c^  =  cos  $  sin  ^,  ^3  =  cos  6. 

338.   It  is  evident,  geometrically,  from  (8),  that  we  must  have 

(11) 


For  just  as  the  first  of  the  equations  (7)  expresses  that  the  sum 
of  the  projections  on  Ox  of  the  co-ordinates  x',y't  z1  is  equal  to 
x,  so  the  first  of  the  equations  (11)  expresses  the  equality  of  x' 
to  the  sum  of  the  projections  of  x,  y,  z  on  the  axis  Ox'\  and 
similarly  for  the  other  equations. 

Now  the  solution  of   the  equations  (7)  for  xft  y',  z1  should 
give  the  values  (11).     Putting 

a\    ai    <*s 

*i     ^2     ^3      =A, 


KINETICS   OF   A   RIGID    BODY.  [339. 

solving  the  equations  (7)  for  x\  and  comparing  the  coefficients 
of  x,  y,  z  to  those  in  (i  i),  we  find  the  following  relations  : 


(12) 


339.  Squaring  and  adding  these  equations  and  applying  the 
relations  (9),  we  find  after  reduction 

A2=i. 

The  two  values  of  A,  + 1  and  —  i,  correspond  to  the  two 
different  relations  between  the  two  rectangular  systems,  which 
might  perhaps  be  called  like  and  unlike.  Two  systems  are  alike 
if  their  positive  axes  can  be  brought  to  coincidence ;  tjjey  are 
unlike  if  this  cannot  be  done.  It  is,  of  course,  always  possible 
to  bring  the  axes  Ox'  and  Oy'  to  coincidence  witli  Ox  and  Oy, 
respectively.  But  after  having  accomplished  this,  the  axis  Ozf 
may  fall  along  Oz,  in  which  case  the  systems  are  alike,  or  it 
may  fall  into  the  opposite  direction,  when  the  systems  are 
unlike. 

Now  if  Ox1  coincides  with  Ox,  Oy1  with  Oy,  we  have  a^  =  i , 
^2=  i  ;  and  c%=  +  i  for  like  systems,  cz=  —  i  for  unlike  systems  , 
as  the  other  6  cosines  are  zero,  we  find  that  A=  -f  i  corresponds 
to  like  systems,  and  A=  — i  to  unlike  systems.  For  it  is  evi- 
dent that  the  motion  of  one  system  with  respect  to  the  other 
cannot  affect  A  so  as  to  change  from  one  of  these  values  to  the 
other. 

In  mechanics  the  systems  should  generally  be  such  that  they 
can  be  brought  to  coincidence.  We  assume  therefore  A=  i. 

With  this  value  of  A,  the  equations  (12)  and  the  similar  rela- 
tions obtained  by  cyclical  permutation  of  the  subscripts  give 
the  identities  : 


340.]  BODY   WITH    FIXED   POINT. 

340.  If  the  axes  Ox\  Oy\  Oz1  be  the  principal  axes  at  O,  the 
equations  (7)  exhibit  the  relations  between  the  system  of  the 
principal  axes  and  a  fixed  system  with  the  same  origin  O,  by 
means  of  the  cosines  of  the  9  angles  between  the  axes  of  the 
two  systems.  They  can  be  used  to  derive  Euler's  equations  by 
.a  purely  analytical  process  from  the  equations  (7),  Art.  224. 

To  accomplish  this  we  must  form  the  quantities  ^m(yz—zy}, 
^m(zx—xz},  ^m(xy—yx\  We  need  therefore  x,  j/,  z.  Now, 
differentiating  the  expressions  (7)  with  respect  to  the  time  and 
remembering  that  x't  /,  z'  are  independent  of  the  time,  we 
find: 


Czz'.  (14) 

To  introduce  the  angular  velocities  tolt  co2,  o>3  about  the  prin- 
cipal axes  at  O,  we  observe  that  the  direction-cosines  alt  a2>  as 
of  the  axis  Ox  can  be  regarded  as  the  co-ordinates  of  the  point 
situated  on  Ox  at  unit  distance  from  O.  The  components  of 
the  linear  velocity  of  this  point,  arising  from  w^  o>2,  ®3  are 

al  =  <22a>3  —  #  3o>2,     tf  2  =  a&>\  ~~  ^1^3*     ^3  =  ^1^2  7  a<iF>\  J 

and  similarly  we  have  for  points  at  unit  distance  from  O  on  Oy 
.and  Oz\ 


It  should  be  noticed  that  the  motion  of  a  body  with  a  fixed 
point  is  fully  determined  by  the  motion  of  two  of  its  points,  not 
in  the  same  line  with  the  fixed  point  ;  the  third  point  is  here 
only  introduced  to  preserve  the  symmetry. 

Substituting  these  values  in  (14),  we  find 


X  = 

y  =  (£2a>3  -  £3o>2X  +  (^ft?!  -  ^wj})/  +  (bi&i  —  b#>^z\        (15) 


!88  KINETICS   OF   A   RIGID    BODY.  [341. 

341.  From  (7)  and  (15)  we  now  find,  if  we  remember  that 
2my'z'  =  o,  2mz'x'  =  o,  ^m^y  =  ot  since  Ox\  Oy\  Oz'  are  prin- 
cipal axes  at  O  : 

2m  (y'z  —  zy)=-  (b2cz  —  2 


or    applying    the    relations    (13)    and    denoting    the    principal 
moments  of  inertia  at  O  by  Iv  72,  /3  : 

*%m(yz  —  zy)  =  alllco1  -f-  #  2/2ft>2  +  #3/30)3. 

The  quantities  *2m(zx—X!5)  and  ^m(xy—yx)  are  obtained  from 
this  result  by  cyclical  permutation  of  the  letters  a,  b>  c. 
Thus  the  equations  (7),  Art.  224,  assume  the  form  : 

-rX^i/i®!  +  a2f2a)2  +  tf  3/3&>3)  =  H* 
at 


^'  (!  6> 

--  (^x/!®!  4-  ^2®2  +  ^B^B)  =  H* 
at 

0 

The  geometrical  meaning  of  these  equations  is  apparent. 
/i®!,  /2w2,  /3w3  are  the  components  along  the  principal  axes  of 
the  vector  of  the  resultant  impulsive  couple  H  (Art.  319); 
hence  <21/1ft)1-f-^2/2ft)24-^3/3a)3  is  the  component  of  H  along  the 
fixed  axis  Ox\  the  equations  (16)  express  therefore  the  fact 
that  H  is  geometrically  equal  to  the  geometrical  derivative 
of  H  with  respect  to  the  time  (see  Art.  327)  ;  they  can  be 
written  in  the  form 

dHz     TT 


342.  If  the  equations  (16)  be  multiplied  first  by  al9  dv  clt 
then  by  a2,  £2,  <:2,  finally  by  ^3,  £3,  c3,  and  each  time  added,  the 
right-hand  members  of  the  resulting  equations  will  evidently 
represent  ffv  '  Hv  ffs,  respectively,  i.e.  the  components  of  H 


343-] 


BODY   WITH   FIXED   POINT. 


I89 


along  the  principal  axes  at  O,  The  left-hand  members  reduce 
also  to  a  simple  form  if  the  differentiations  indicated  in  (16) 
be  performed,  the  values  for  alt  #2,  a3,  b^  •••  be  substituted 
from  Art.  340,  and  the  relations  (9)  be  applied.  As  final  result 
we  find  Euler's  equations  : 


»  3  +  (72  -  /! 


3.     CONTINUOUS    MOTION   WITHOUT   FORCES. 

343.  In  the  particular  case  when  no  external  forces  are  act- 
ing, the  motion  of  a  rigid  body  about  a  fixed  point  admits  of  an 
elegant  geometrical  interpretation  which  is  due  to  Poinsot. 

As  there  are  no  external  forces,  we  have  //"=o,  and  hence 
H  is  constant  in  magnitude  and  direction.  The  plane  of  this 
couple  is  the  invariable  plane  (see  Arts.  230-232)  which  always 
exists  in  the  case  of  no  forces ;  its  vector  indicates  the  invari- 
able direction. 

The  body  can  be  replaced  by  its  momental  ellipsoid  at  O,  and 
the  invariable  plane  can  be  imagined  placed  so  as  to  be  tan- 
gent to  this  ellipsoid  at  a  point  P' 
(Fig.  43).  The  radius  vector  OP'  =  p' 
of  the  point  of  contact  P'  is  the  diam- 
eter of  the  ellipsoid  conjugate  to  the 
invariable  plane ;  hence  the  line  OP' 
is  the  instantaneous  axis  /  of  the 
rotation  (Art.  321). 

Now  it  can  be  shown  that  the  per- 
pendicular distance  q1  of  O  from  the 
invariable  plane  (this  plane  being 
always  placed  so  as  to  be  tangent 
to  the  varying  positions  of  the  mo- 
mental  ellipsoid)  is  constant ;  it  then 
follows  at  once  that  the  motion  of  the  body  consists  in  the  rolling 
of  its  momental  ellipsoid  over  the  invariable  plane. 


190  KINETICS   OF   A   RIGID    BODY.  [344, 

344.   To  prove  that  q'  is  constant,  it  should  be  remembered 
that,  by  (7),  Art.  321,  we  have 

'-"•?•;&•  \ 

As  H  is  constant  in  our  case,  it  only  remains  to  show  that  o>/pr 
is  constant.  This  follows  from  the  expression  (9)  for  the  kinetic 
energy  Tt  given  in  Art.  322,  viz. 


for,  as  there  are  no  external  forces,  no  work  is  done,  and  the 
kinetic  energy  must  remain  constant  ;  hence  co/p'  is  constant, 
and  co  is  directly  proportional  to  pf. 

Moreover,  the  expression  (14)  of  Art.  322  shows  that 


a)  cos  <f)  —  ~=  const.,  (i) 

H 

that  is,  the  projection  a>  cos<£  of  the  angular  velocity  co  on  the 
invariable  direction  remains  the  same  throughout  the  motion. 

345.  It  has  been  pointed  out  in  Part  I.,  Art.  35,  that  the 
motion  of  a  rigid  body  with  a  fixed  point  can  always  be 
regarded  as  produced  by  the  rolling  of  the  cone  of  the  body 
axes  over  the  cone  of  the  space  axes,  these  cones  having  their 
common  vertex  at  the  fixed  point  O.  The  body  axes,  i.e.  the 
lines  /'  of  the  body  that  become  instantaneous  axes  of  rotation 
in  the  course  of  the  motion,  form  a  cone,  invariably  connected 
with  the  momental  ellipsoid  at  <9,  and  intersecting  this  ellip- 
soid in  a  curve  fixed  in  the  body.  This  curve  has  been  called 
by  Poinsot  the  polhode  (or  path  of  the  instantaneous  pole  P\ 

Fig.  43). 

The  cone  of  the  space  axes  /,  which  is  fixed  in  space,  inter- 
sects the  invariable  plane  in  a  curve  called  herpolhode  (or  creep- 
ing path  of  the  pole).  During  the  motion  of  the  body,  the 
polhode  rolls  over  the  herpolhode. 


347-]  BODY   WITH   FIXED   POINT.  !9! 

346.  The  equations  of  the  polhode  are  easily  obtained  by 
considering  that  this  curve  is  the  locus  of  those  points  of  the 
ellipsoid  whose  tangent  plane  has  the  constant  distance  q*  from 
the  centre  O.  Hence,  denoting  the  semi-axes  of  the  momental 
ellipsoid  by  a,  b,  c,  the  equations  of  the  polhode  are 

2          2       .2  2          2       «2 


It  can,  therefore,  be  regarded  as  the  intersection  of  the  momen- 
tal ellipsoid  with  a  coaxial  ellipsoid  whose  semi-axes  are  a*/q', 


Multiplying  the  second  equation  by  q®,  and  subtracting  the 
result  from  the  first  equation,  we  find  the  equation  of  the  cone 
of  the  body  axes 


This  is  a  cone  of  the  second  order,  concentric  and  coaxial  with 
the  momental  ellipsoid. 

The  polhode  evidently  consists  of  two  equal  separate  branches, 
of  which  it  is  sufficient  to  consider  one.  Each  branch  has  four 
vertices  situated  in  the  principal  planes  of  the  ellipsoid. 

The  herpolhode  is  confined  between  two  concentric  circles 
whose  centre,  is  the  projection  of  O  on  the  invariable  plane.  It 
is  a  transcendental  curve  and  is  in  general  not  closed. 

347.  If  the  momental  ellipsoid  is  an  ellipsoid  of  revolution,  the 
polhode  consists  of  two  circles,  and  the  herpolhode  is  also  a  circle ;  as 
>'  is  in  this  case  constant,  it  follows  that  <o  remains  constant. 

If  we  assume  in  the  general  case  a  >  b  >  c,  the  polhode  reduces  to  two 
points  whenever  q1  ==  a  or  ql  =  c.  The  rotation  then  takes  place  about 
a  principal  axis  and  is  permanent.  If  q'  =  b  (which  does  not  necessarily 
mean  that  the  axis  of  rotation  coincides  with  the  middle  axis  b\  the 
cone  of  body  axes  reduces  to  two  planes 


I92  KINETICS   OF   A   RIGID    BODY.  [348 

each  of  which  intersects  the  ellipsoid  in  an  ellipse.  These  ellipses 
divide  the  surface  of  the  ellipsoid  into  two  pairs  of  opposite  regions,  one 
about  the  greatest  axis  a,  the  other  about  the  least  c. 

As  long  as  a  >  <?'  >  b,  the  polhode  lies  in  the  former  region,  and  the 
cone  of  body  axes  has  a  as  its  axis.  If  b  >  <?'  >  c,  the  polhode  lies  in 
the  other  region,  and  c  is  the  axis  of  the  cone. 

Two  polhodes  cannot  intersect ;  for  if  they  did,  the  tangent  plane  a 
the  point  of  intersection  would  have  two  different  distances  from  the 
centre,  which  is  impossible. 

348.  The  motion  of  a  body  is  called  stable  if  after  a  slight  distur 
bance  the  body  tends  to  resume  the  original  motion.     In  our  case 
slight  disturbance  displaces  the  instantaneous  axis  from  one  polhode  to 
another  near  by.     Hence  if  the  polhode  be  situated  very  near  to  one 
of  the  bounding  ellipses,  the  motion  is  not  stable,  because  a  slight  dis 
turbance  might  change  the  polhode  to  one  in  the  other  region.     The 
motion  is  therefore  the  more  stable  the  more  closely  the  polhode  sur 
rounds  either  the  greatest  or  the  least  axis  of  the  ellipsoid. 

If,  however,  one  of  the  regions  between  the  ellipses  be  very  narrow 
which  will  be  the  case  if  two  of  the  axes  of  the   ellipsoid  are  nearly 
equal,  a  polhode  in  this  region,  though  close  to  the  vertex,  may  stil 
approach  very  near  to  the  ellipses  so  as  to  make  the  motion  unstable. 

349.  Integration  of    Euler's  Equations.      As    H=o,    Euler's 
equations  (4),  Art.  328,  are 


(4 

ai 

Multiplying  by  oj,  o>2,  a>3,  and  adding,  we  find 


whence,  by  (17),  Art  323, 

This  is  nothing  but  the  equation  of  kinetic  energy. 


35I-]  BODY   WITH   FIXED   POINT. 


193 


Again,  multiplying  the  equations  (4)  by  T^,  /2a>2,  73a>3,  and 
adding,  we  find  similarly,  by  (15),  Art.  323, 

A  V  +  A  V  +  /8  V=#*  =  const.  (6) 

This  is  the  principle  of  areas  or  of  the  invariable  plane. 
As,  moreover, 

&)12  +  ft)22  +  ft)32  =  w2,  (7) 

we  have  three  equations  (5),  (6),  (7)  for  determining  a^2,  a>22,  a>32. 
Their  solution  gives,  after  some  reductions, 


_  _ 

-- 


(A-AXA-A)  (A-AXA-A) 

^ 


350.    To  find  the  time,  multiply  the  equations  (4)  by 
<w2/A>  o)3//3,  and  add.     This  gives 


°r 


^1      2  (/2-/3)(/3-/l)(A-/2) 

«  =  -  M  -  -  ' 


In  this  equation  the  values  (8)  should  be  substituted  for  ®j,  o)2> 
<»3.     For  the  sake  of  brevity,  let  us  put 


we  then  find 


-  a>2)  (a>2  -  72) 

This  is  an  elliptical  integral  whose  discussion  is  beyond  the 
scope  of  the  present  treatise. 

351.  It  remains  to  determine  the  position  of  the  moving 
system  formed  by  the  principal  axes,  with  respect  to  a  fixed 
.system  of  axes  through  O,  by  means  of  Euler's  angles  0,  <£,  ty 
(Art.  333).  After  finding  co  as  a  function  of  t  from  (9),  we 
have,  by  (8),  colf  a>2,  &>3  as  functions  of  t.  Substituting  these 

PART  III  —  13 


KINETICS   OF   A   RIGID   BODY.  [352. 

values  into  the  equations  (5)  or  (6),  Art.  335,  we  have  a  system 
of  differential  equations  of  the  first  order  whose  integration 
gives  0,  $,  T/T  as  functions  of  t. 

352.  To  illustrate  the  method  by  a  simple  example,  let  us  consider 
the  case  of  a  body  whose  momental  ellipsoid  is  an  ellipsoid  of  revolution. 

Let  /!  =  /2;  then,  putting  (/2  —  /3)//!=  -  (78  —  /i)//2  =  A,  Euler's. 
equations  (4)  become 


o,     -f=o.  (10) 

dt  dt  dt 

The  last  of  these  equations  shows  that  the  component  of  the  angular 
velocity  about  the  axis  of  revolution  of  the  body  is  constant.  The 
other  two  equations  give 


whence  cuj2  +  a>22  =  const.  =  <o02,  (i  i 

where  w0  denotes  the  constant  angular  velocity  about  the  projection  o 
the  instantaneous  axis  on  the  equatorial  plane  of  the  body.  The  result 
ing  angular  velocity  is,  therefore,  constant,  viz. 

to  =  V  o>02  +  <i)32.  (  1  2 

353.  The  inclination  of  the  instantaneous  axis  to  the  principal  axe 
a,  b  varies,  but  its  inclination  to  the  axis  c  is  constant,  viz.  =  cos~l(o)3/a)0) 
The  cone  of  the  body  axes  is,  therefore,  a  cone  of  revolution  about  th( 
axis  cy  and  the  polhode  is  a  circle.     The  herpolhode  is,  therefore,  like 
wise  a  circle,  and  the  space  axes  form  a  cone  of  revolution  (comp.  Art 
347).     As  the  two  cones  are  always  in  contact  along  the  instantaneou 
axis,  this  axis  lies  in  the  same  plane  with  the  vector  H  and  the  axis  o 
revolution  of  the  body. 

354.  To  find  the  angular  velocities  wj,  o>2  as  functions  of  the  time 
differentiate  the  first  of  the  equations  (10),  and  eliminate  d^/dt  wit! 
the  aid  of  the  second.    This  gives 

^2<ui  i  \2    2 

-^  +  A  Va^  =  o, 

whence  o^  =  C\  cos  Ao>3/  -f  C2  sin  Aa>3/.  (13" 

The  other  component,  o>2,  can  now  be  found  from  the  first  of  the  equation 
(10): 

<i>2  =  -  --  ^  =  —  Cj  sin  Aw3/  +  C2  cos  A<o3/.  (14' 

Aoo3  dt 


356.]  BODY   WITH    FIXED   POINT.  !95 

To  determine  the  constants  Ci,  C.2)  the  initial  values  of  o^,  <o2,  say  at 
the  time  /=  o,  should  be  known.  Let  e  be  the  angle  made  at  this  time 
by  a>0  with  the  principal  axis  b.  Then  the  initial  values  of  o^,  o>2  are 
w0sine,  eo0  cos  e,  and  the  substitution  of  /=  o  in  (13)  and  (14)  shows 
that  Cj  =  w0  sin  e,  C2  =  w0  cos  e.  Hence  we  have,  finally, 

coj  =  w0  sin  (Awg/1  +  e)  ,   o>2  =  o>0  cos  (A.<u3/+  e),  o>3  =  const.     (15) 


355.  To  determine  the  position  of  the  body  at  any  time  /  with 
respect  to  fixed  axes  through  <9,  let  us  take  as  axis  of  z  the  fixed 
direction  of  H,  which  is  perpendicular  to  the  invariable  plane.  The 
cosines  of  the  angles  made  by  this  axis  with  the  principal  axes  are 
found  similarly  as  in  Art.  335  (see  Fig.  42)  : 

cos  ZX'=  sin  <£  sin  0,    cos  ZY'  =  cos  <£  sin  0,    cosZZ'=cosO. 

Hence  the  components  ff1=71a>lf  H2=I2w2,  J^3=I3(a3  of  H  along  the 
principal  axes  are 

/id)!  =7/sin  <£  sin  0,  72o>2  =ZTcos  <f>  sin  0,  ^wg  =^ffcos  6. 
These  equations  give 

cos0=^,  (16) 

and,  as  7j  =/2,  tan  </>  =  —  =  tan  (A<o3/+e), 

o>2 


by  (15)  ;  hence  <#>  =  X^t  +  ^  (17) 

where  ^>0  is  the  value  of  <£  for  /  =  o.     Thus  it  appears  that  the  angle  6  is 
constant,  while  <£  increases  proportionally  to  the  time. 

356.  To  find  ^,  we  may  use  the  third  of  the  equations  (5),  Art.  335, 
viz.  o)3  =  <£  +  \j/  cos  0.     As  cos  0  =  1^/11,  <j>  —  A<o3,    A  =  (7j 
we  find 


whence 


dt 

TT 


It  appears  then  that  the  equatorial  plane  X'  Y1  of  the  body  remains 
at  a  constant  inclination  0  to  the  invariable  plane,  while  the  nodal  line 
ON  (Fig.  42)  turns  uniformly  in  this  invariable  plane  and  a  radius  of 
the  body  in  the  equatorial  plane  turns  also  uniformly  in  the  equatorial 
plane. 


196  KINETICS   OF  A   RIGID   BODY.  [357. 

V.    Free  Rigid  Body. 

I.     INITIAL    MOTION    DUE    TO    IMPULSES. 

357.  Kinematically,  the  most  general  motion  of  a  rigid  body 
consists,  at  every  instant,  of  a  twist,  or  screw-motion  about  a 
certain  line,  called  the  instantaneous  axis  1 ;   that  is,  the  body 
has,  for  an  element  of  time,  an  angular  velocity  o>  about  /  and 
at  the  same  time  a  velocity  of  translation  v  along  this  axis  (see 
Part  I.,  Arts.  43,  44,  294).      During  the  next  element  of  time 
the  body  will,  in  general,  rotate  about  a  different  axis  with  a 
different  angular  velocity  and  will  have  a  different  linear  velocity 
along  the  new  axis. 

358.  It  has  also  been  shown  in  kinematics  (Part  I.,  Art.  257) 
that  the  angular  velocity  &  about  /  can  be  replaced  by  an  equal 
angular  velocity  about  any  parallel  axis  /',  in  connection  with  a 

certain  velocity  of  translation.  For  without 
changing  the  state  of  motion  we  can  give  the 
body  two  equal  and  opposite  rotations  about 
/' ;  i.e.  we  can  introduce  along  /'  (Fig.  44)  two 
equal  and  opposite  rotors  a>,  —  o> ;  and  —  o> 
about  P  combines  with  o>  about  /  to  a  rotor 
couple,  which  is  equivalent  to  a  velocity  of 
translation  /o>,  perpendicular  to  the  plane 
(/,  /'),  p=OO1  being  the  distance  of  the  par- 
allel axes.  The  velocity  of  translation  p& 
can  now  be  combined  with  v  to  a  resultant  velocity  of  transla- 
tion v1  =  V^2  +/2o>2  inclined  to  F  at  an  angle  ^>  =  tan~1  (pco/v). 

It  thus  appears  that  the  instantaneous  motion  of  a  free  rigid 
body  can  be  regarded  as  a  rotation  about  any  line  parallel  to  the 
instantaneous  axis,  in  combination  with  a  certain  velocity  of 
translation  inclined  to  this  line. 

On  account  of  the  dynamical  properties  of  the  centroid  of  a 
rigid  body,  it  will  generally  be  found  convenient  to  select  the 


361.] 


INITIAL   MOTION    OF   FREE   BODY. 


197 


axis  of  rotation  so  as  to  pass  through  the  centroid  ;    we  shall 
then  call  it  the  centroidal  instantaneous  axis  I. 

359.  Dynamically,  the  instantaneous  motion  of  a  free  rigid 
body  is  determined  by  the  momenta  of  its  particles.     These 
momenta  can  be  reduced,  for  any  point   O  as   origin,  to  a  re- 
sultant momentum  and  a  resultant  couple,  or  angular  momen- 
tum, and  these  can  be  regarded  as  due  to  a  certain  system  of 
impulses.     This  reduction  will  at  the  same  time  lead  to  the 
solution  of  the  converse  problem,  viz.  to  determine  the  initial 
motion  produced  by  a  system  of  impulses  acting  on  a  rigid  body 
at  rest,  and  the  change  in  the  instantaneous  motion  due  to  such 
a  system  when  the  body  is  not  at  rest. 

360.  Translation.     The  velocities  u  of  all  points  being  equal 
and  parallel  in  the  case  of  translation,  the  momenta  mu  of  all 
particles  are  parallel  and  have  (see  Arts.  6-8)  a  single  resultant 


passing  through  the  centroid  G  of  the  body.  If  the  whole 
mass  M  be  regarded  as  concentrated  at  the  centroid,  Mu  is  the 
momentum  of  the  centroid.  This  momentum  can  be  produced 
by  applying  at  the  centroid  a  single  impulse  R=Mu.  Hence 
to  impart  to  a  free  rigid  body  of 
mass  M  a  velocity  of  translation 
u,  it  is  sufficient  to  apply  at  the 
centroid  an  impulse  R  =  Mu. 


361.   Rotation.      Let   us  take 
the  instantaneous  axis  /  as  axis 
of  £,  and  the  axis  of  x  so  as  to 
pass    through    the    centroid    G 
(Fig.  45).    The  momentum  mar      Hx 
of  any  particle   of   mass   m,  at      /£ 
the   distance  r  from  /,  has  the 
components  —may,  mwx,  o;   and  as 


z 

I 

Hz 

\ 

\ 

\ 

\H 

O 

/    \           Htf     R 

\                /               V 

/ 

\     i        / 

G/x 

\    i       / 
\  i     / 

/ 

\i  / 

/  . 

fr 

Fig.  45. 


the 


KINETICS   OF   A   RIGID   BODY.  [362. 

resultant  momentum  has  the  components  o,  Max,  o  ;  it  is 
therefore  perpendicular  to  the  plane  through  axis  and  centroid. 
Hence  the  resultant  impulse  R  at  O  must  be  equal  in  magnitude 
and  direction  to  the  momentum  Mv  =  Me»x  of  the  centroid,  due 
to  the  rotation  CD  about  the  instantaneous  axis  1. 

The  resultant  angular  momentum  is  found  just  as  in  Art.  317  ; 


it  is  =  o>V£'2  +  Z?2-f-J£%2  and  has  the  components  —  Eco  along  Ox, 
—  Deo  along  Oy,  and  Ceo  along  the  instantaneous  axis  Oz. 

It  follows  that  a  pure  rotation  of  angular  velocity  co  about 
an  axis  1  can  be  imparted  to  a  free  rigid  body  by  the  combined 
action  of  an  impulse  R  and  an  impulsive  couple  H.  The  im- 
pulse R  =  M(ox  is  perpendicular  to  the  plane  (/,  G),  and  passes 
through  the  foot  O  of  the  perpendicular  let  fall  from  the 
centroid  G  on  the  axis;  it  vanishes  only  when  x=  OG  =  o;  i.e. 
when  the  instantaneous  axis  passes  through  the  centroid.  The 
remarks  of  Art.  318  apply  to  the  couple  without  change. 

362.    As  mentioned   above,  it  is  often  more  convenient  to 
take  the  centroid  G  as  origin  for  the  reduction  of  the  impulses. 
To  reduce  the  system  of   impulses  R,  H,  determined   in    the  j 
preceding  article,  to  G  as  origin  and  to  parallel  axes  (Fig.  46), 

it  is  only  necessary  to  apply  R  \ 
and   —  R  at  G\   we  then   have 
the  resultant  impulse  R  =  Mu>x 
at  G,  and  the  couple  formed  by 
/  R  at   O,  and   -R  at   G.     The 

-  *  -  1     moment  of  this  couple  is  —  Rx 
=  —Mo)xz  ;  its  vector  is  parallel 

,^  -  ->  -  ^  -  -,  to  the  instantaneous  axis  /,  and 

*      /  '  can    therefore    be   added   alge- 

*/  .  braically  to  Hn,  while  Hx  and  Hy  \ 

remain   unchanged.     Thus   the  j 

components  of  the  resulting   couple   for  the   reduction  to  the  j 
centroid     are     Hx=-E<*y    Hy=-Da>,   Hz=(C-M^2)oo=Coof 
where  Cf  =  C—Mx2  is   the   moment  of  inertia  about  the  cen-  \ 


364-]  INITIAL   MOTION    OF   FREE   BODY. 


199 


troidal  axis  /  whose  distance  from  /  is  x  (see  Art.  250),  while 
D  and  E  are  the  products  of  inertia  for  the  new  co-ordinate 
planes  through  G. 

These  results  can,  of  course,  also  be  derived  directly  by 
reducing  the  momenta  for  G  as  origin,  the  centroidal  instanta- 
neous axis  /  as  axis  of  z,  and  the  plane  through  G  and  the 
instantaneous  axis  /  as  the  ^^- 


363.  It  thus  appears  that  the  form  of  the  results  for  this  new 
system  of  co-ordinates  is  exactly  the  same  as  in  Art.  354;  but 
C,  D,  E   refer  now  to  the  new  co-ordinate   axes   and   planes. 
Hence   a  pure  rotation  about  any  instantaneous  axis  1,  at  the 
distance  x  from  the  centroid  G,  can  be  produced  by  an  impulse 
R  and  a  couple  H,  the  impulse  R  being  equal  to  the  momentum 
Mcox  of  the  centroid)  due  to  the  rotation,  and  passing  through  G 
at  right  angles  to  the  plane  (1,  G),  while  the  vector  of  the  couple 
H    has   in   general    three    components    Hx=  —  Eo>,    Hy=—  D&>, 
Hz=Co>. 

The  geometrical  relation  between  the  vector  H  and  the  rotor 
o)  can  again  be  illustrated  by  means  of  the  ellipsoids  of  inertia, 
as  in  Arts.  320,  321.  The  developments  of  these  articles  apply 
without  change  if  the  foot  O  of  the  perpendicular  let  fall  from 
the  centroid  on  the  instantaneous  axis  /0  be  substituted  for  the 
fixed  point  ;  they  apply  likewise  if  the  centroid  G  be.  substituted 
for  the  fixed  point,  in  which  case  the  momental  ellipsoid  be- 
comes the  central,  and  the  reciprocal,  the  fundamental  ellipsoid. 

364.  The  resulting  impulse  R  =  Mo*x  vanishes  only  for  x=o\ 
i.e.  when  the  instantaneous  axis  /  passes  through  the  centroid. 
In  other  words,  pure  rotation  about  an  axis  not  passing  through 
the  centroid  cannot  be  produced  by  an  impulsive  couple  alone. 

On  the  other  hand,  pure  rotation  about  a  centroidal  axis  can 
always  be  regarded  as  due  to  an  impulsive  couple  alone  ;  and 
conversely,  the  effect  of  a  single  impulsive  couple  on  a  free  rigid 
body  is  to  produce  pure  rotation  about  a  centroidal  axis.  But  it 


200  KINETICS    OF   A   RIGID   BODY.  [365. 

should  always  be  remembered  (see  Art.  318)  that  the  axis  of 
rotation  /  is  parallel  to  the  vector  H  of  the  couple  only,  and 
always,  if  D=o  and  E=o;  i.e.  if  the  vector//  is  parallel  to 
a  principal  axis  at  G.  Hence  pure  rotation  about  a  centroidal 
principal  axis  can  be  produced  by  a  single  couple  whose  plane 
is  perpendicular  to  the  axis ;  and  conversely,  a  couple  whose 
plane  is  perpendicular  to  a  centroidal  principal  axis  produces 
pure  rotation  about  this  axis.  The  relation  between  the  mo- 
ment H  of  the  couple  and  the  angular  velocity  o>  produced  is 
in  this  case  H=fcD  =  Ma>g2,  where  /  is  the  moment,  q  the 
radius  of  inertia  for  the  principal  axis. 

365.  To  find  the  condition  under  which  the  system  of  im- 
pulses producing  pure  rotation  may  reduce  to  a  single  impulse 
/?,  we  have  only  to  reduce  the  system  of  impulses  to  its  central 
axis  (comp.  Part  II.,  Arts.  204-206).     For  this  line  which   is 
parallel  to  R  has  the  property  that  if  any  point  on  it  be  taken 
as  origin  of  reduction,  the  couple  has  its  vector  parallel  to/? 
and  has  its  least  value  //0,  which  is  equal  to  the  projection  on 
this  line  (i.e.  on  the  direction  of  R)  of  the  vector  H  for  any 
reduction.     Now,  as  in  our  case  the  components  Hx  and  Hz  are 
both  perpendicular  to  R  (see  Fig.  45),  it  follows  that 

//0= //=-/?«. 

This  vanishes  only  with  the  product  of  inertia  D=^myz. 

Hence  pure  rotation  about  an  instantaneous  axis  1  can  be  pro- 
duced by  a  single  impulse  R  =  Mo>x  only,  and  always,  if  1  is  so 
situated  that  the  product  of  inertia  D  =  2myz  vanishes  for  the 
planes  through  G  and  1  and  through  G  perpendicular  to  1.  In 
particular,  this  is  evidently  the  case  whenever  \  is  a  principal 
axis  at  the  foot  O  of  the  perpendicular  let  fall  on  it  from  the  cen- 
troid.  (Comp.  Arts.  309,  310.) 

366.  It  remains  to  find  the  position  of  the  central  axis,  i.e. 
of  the  line  of  action  of  the  single  impulse  R  capable  of  pro- 


367-] 


INITIAL   MOTION   OF   FREE   BODY. 


201 


ducing  pure  rotation  about  the  instantaneous  axis  /  provided 
it    satisfies    the    condition 

just    mentioned.     This   can  j 

be   done,   if   R   and  H  are  T 

known  for  the  centroid,  by 
transferring  R  to  parallel  po- 
sitions so  as  to  reduce  the 
components  Hn  and  Hx  to 
zero.  Thus,  to  destroy 
Hz=Cco  we  have  only  to 
transfer  R  from  G  along 
the  axis  of  x  to  a  point  O1 
(Fig.  47)  at  a  distance 


Fig.  47. 


'=.^  from  G  such  that  xxl  =  q^J  where  ~x=OG,  and  ^  is  the 
radius  of  inertia  for  the  centroidal  instantaneous  axis  7.  For 
then  the  couple  resulting  from  the  transfer  has  a  vector  along 
the  axis  of  z  equal  to  —  Rx^  —  —  Ma)xxl  =  —  M^  —  —  Hg. 

Next  to  destroy  Hx—  —  Ea>  =  —  a&mzx,  we  transfer  the  point 
of  application  of  R  parallel  to  the  instantaneous  axis  /  to  a  point 
Olt  at  a  distance  O1Ol=zl  from  O',  such  that  Rzl=—Hxt 
whence  z±  =  ^mzx/Mx. 

If  the  point  O1  be  taken  as  origin  of  reduction,  the  system 
of  impulses  reduces  to  R  at  Ov  and  the  couple  //0  =  //.=  —  £)&, 
whose  vector  is  parallel  to  R.  Thus  the  central  axis,  which  has 
of  course  the  direction  of  R,  and  is  therefore  perpendicular  to 
the  plane  (/,  G),  meets  this  plane  at  a  point  Ox  whose  co-ordi- 
nates are  x^  —  q^/'x^  zl  =  E/mx.  It  is  easy  to  see  that  these 
results  agree  with  the  developments  of  Arts.  309,  310,  the 
centre  of  percussion,  if  it  exists,  being  situated  on  the  central 
axis. 


367.  Twist  or  Screw  Motion.  In  the  most  general  case  the 
motion  of  a  rigid  body  consists  of  an  angular  velocity  o>  about 
the  instantaneous  axis  /and  a  simultaneous  velocity  of  transla- 
tion a  along  this  axis  (Art.  357). 


202 


KINETICS   OF   A   RIGID   BODY. 


[368. 


Now,    by   Art.   360,   the   velocity  of   translation,   u,  can  be 
regarded  as  due  to  a  single  impulse  R'  =  Mu,  passing  through 

the  centroid  G  and  parallel  to  u,  i.e.  to 
/  (Fig.  48).  Again,  by  Art.  361,  the 
angular  velocity  co  about  /  can  be  re- 
garded as  due  to  an  impulse  R"=Ma>x 
•^  through  G  at  right  angles  to  the  plane 
(/,  G),  in  combination  with  a  couple 
whose  vector  H  has  the  components 
Hx=  -Eto,  Hy  =  -Deo,  Hz=C<o.  The 
two  impulses  R\  R"  combine  to  form 
a  single  resultant  impulse, 


Fig.  48. 


R 


'2  +  R  "2  = 


inclined  to  /at  an  angle  $  =  tan  1(a&/u).  It  should  be  noticed 
that  the  factor  Vw2+o>2^2  is  the  velocity  v  of  the  centroid  due  to 
the  twist,  so  that  the  resultant  impulse  R  =  M  v  is  equal  to  the  mo- 
mentum of  the  centroid.  The  resultant  couple  H=o)^/C2 
is  the  same  as  in  the  case  of  pure  rotation. 

368.  The  problem  of  determining  the  initial  motion  produced 
in  a  free  rigid  body  at  rest  by  a  given  system  of  impulses  finds 
its  geometrical  solution  in  the  preceding  articles.  It  should 
also  be  remembered  that  the  motion  about  the  centroid  takes 
place  as  if  the  centroid  were  fixed  so  that  all  the  developments 
of  Arts.  313-323  can  be  applied  by  substituting  the  centroid  G 
for  the  fixed  point  O. 

It  will  generally  be  best  to  reduce  the  given  impulses  to  a 
resultant  R,  passing  through  the  centroid,  and  to  a  couple  H. 
By  Art.  360,  the  impulse  R  at  G  produces  a  velocity  of  trans- 
lation, ^> 


By  Arts.  361,  363,  320,  321,  the  couple  //produces  an  angular 

velocity, 

/2\ 


369.] 


INITIAL   MOTION    OF   FREE   BODY. 


203 


about  a  centroidal  axis  /  whose  direction  is  conjugate  to  the 
plane  of  the  couple  H  in  the  central  ellipsoid,  while  a  plane 
perpendicular  to  /is  conjugate  to  the  direction  of  the  vector  H 
in  the  fundamental  ellipsoid.  The  components  of  co  along  the 
principal  centroidal  axes  are  given  by  (5),  Art.  319,  viz.  : 


ucos  a 


u. 


usina 


l 

ucos  a 
u 

^ 

O 


where  Iv  72,  73  are  the  principal  centroidal  moments  of  inertia 
and  7/j,  J72,  7/3  the  components 
of  H  along  the  centroidal  prin- 
cipal axes. 

The  direction  of  the  instan- 
taneous axis  having  thus  been 
determined,  its  position  can  be 
found  by   resolving   u   into   a  usina 
component  u  cos  a  along  /  and 
a   component    u  sin  a   perpen- 
dicular  to   7  (Fig.   49).      The  FiS- 49- 
latter  component  combines  with  co  about  /  to  an  equal  angular 
velocity  co  about  an  axis /parallel  to  /at  the  distance  x—  —  usma/co 
from  7 

The  initial  motion  produced  by  the  impulses  consists,  there- 
fore, of  the  angular  velocity  co  about  /,  and  the  linear  velocity 
These  together  constitute  the  resulting  twist. 


u  cos  a  along  / 


369.   Exercises.* 

(i)  A  homogeneous  straight  rod  AB=  2  a  (Fig.  50)  is  acted  upon 
by  an  impulse  F,  at  the  distance  c  from  the  centroid  G,  at  an  angle  a 
with  the  rod.  Determine  the  initial  motion. 

The  reduction  to  the  centroid  G  gives  the  impulse  F  at  G,  which 
produces  a  velocity  of  translation  u  =  F/M  in  the  direction  of  F,  and 
the  couple  H  formed  by  F  at  C  and  -  F  at  G.  The  moment  of  this 
couple  is  ^sina,  and  its  vector^  is  at  right  angles  to  the  plane  deter- 

*  Most  of  these  problems,  as  well  as  the  discussions  of  Arts.  357-368,  are  adapted 
from  W.  SCHELL,  Theorie  der  Bewegung  und  der  Krafte,  Vol.  II.,  pp.  352-386. 


204 


KINETICS   OF   A   RIGID    BODY. 


[369- 


mined  by  AB  and  F.  As  any  perpendicular  to  the  rod  is  a  principal 
axis,  the  radius  of  inertia  at  G  being  ^  =  Vg-#,  the  couple  Fcsma 
Q  produces  pure  rotation  about  an 

3  /**•      >X-      axis /through  G  at  right  angles  to 

\Q/\/        _,     the  plane   (AB,  F}  (see  Art.  364), 
B    and  we  have 

H  _  3  Fc  sin  a 
'j/tf2 


Fig.  50. 

As  the  axis  7  of  this  rotation  is  perpendicular  to  the  direction  of  the 
velocity  u  of  translation,  their  combined  effect  is  a  pure  rotation  of  the 
same  angular  velocity  <o  about  a  parallel  axis  /  whose  position  is  found 
as  follows  (Art.  368)  :  Draw  through  G,  in  the  plane  (AB,  F),  a  per- 
pendicular to  F,  and  on  this  perpendicular  lay  off 
With  the  values  of  u  and  <o  given  above,  we  have 


3  c  sin  a 

which  can  easily  be  constructed  geometrically.  The  parallel  to  7 
through  O  is  the  instantaneous  axis  about  which  the  rod  begins  to 
rotate  with  the  angular  velocity  o>. 

In  what  direction  and  at  what  distance  from  G  must  the  rod  be 
struck  if  it  is  to  rotate  about  a  perpendicular  through  the  end  A  ? 

(2)  A  homogeneous  plane  lamina  of  mass  M  receives  an  impulse  F  in 
its  plane,  the  distance  of  the  centroid  from  the  direction  of  F  being  Xj. 
Determine  the  initial  motion. 

The  reduction  to  the  centroid  G  (Fig.  51)  gives  F  at  G,  and  a 
couple  .//=  Fxl  whose  vector  is  parallel 
to  a  principal  centroidal  axis.  The  couple 
produces,  therefore,  rotation  about  the  per- 
pendicular /through  G  to  the  plane  of  the 
lamina,  and  this  rotation  combines  with 
the  translation  due  to  F  to  a  single  rotation 
about  a  parallel  axis  /.  To  find  the  position 
of  /,  lay  off  on  the  direction  of  F,  drawn 
through  G,  a  length  GK  equal  to  the  radius 


0' 


-F 


Fig.  51. 


of  inertia  for  /;  join  K  to  the  foot  O1  of  the  perpendicular  let  fall  from 
G  on  F,  and  draw  KO  at  right  angles  to  KO*.  The  instantaneous 
axis  /passes  through  O,  since  GO  •  GO1  =  q*.  The  angular  velocity  is 
o)  =  Fx^Mf. 


INITIAL   MOTION   OF   FREE   BODY. 


205 


— F 


(3)  In  Ex.   (2),  if  the  lamina  be  an  ellipse  of  semi-axes  a,  Ja,  at 
what  point  of  the  major  axis  must  it  be  struck  at  right  angles  to  this 
axis  in  order  to  rotate  initially  about  a  focus  ? 

(4)  A  rigid  body  of  mass  M  receives  an  impulse  F  parallel  to  the 
principal  axis  Gy,  and  meeting  the  principal  axis  Gx  at  the  distance 

GO'  =  Xx  from  the  centroid.     Determine  the  initial  motion  (Fig.  52). 

The  impulse  F  at  O'  is  equivalent  to  an  equal  and  parallel  impulse 
F  through  the  centroid  G,  in  combination  with  the  couple  H '•=.  Fx^ 
whose  vector  is  parallel  to  the  principal 
axis  Gz.  This  couple  produces,  there- 
fore, rotation  about  the  centroidal  prin- 
cipal axis  Gz,  or  7,  of  angular  velocity 
<o  =  Fxi/Mq%,  where  qz  is  the  radius  of 
inertia  for  /.  The  instantaneous  axis  / 
is  parallel  to  7,  and  meets  the  axis  Gx  at 
a  point  O  such  that  GO-  GO1  =  ?32,  or 
putting  GO  =  x,  GO'  =  x1}  such  that 
jcxi  =  q£  j  it  can  be  constructed  as  in 
.Ex.  (2).  v  Fig.  52. 

(5)  Determine  the  impulse  "F  imparted  to  a  body  of  mass  M  when 
struck  at  the  point  O'  (Fig.  53)  of  a  principal  axis  Gx  by  a  particle  of 
mass  m  moving  with  a  velocity  u  parallel  to  another  principal  axis  Gy. 
The  impact  is  assumed  to  be  inelastic  (compare  Art.  305). 

It  has  been  shown  in  Art.  19  that  if  a  particle  of  mass  m  moving  with 
a  velocity  u  impinges  upon  a  particle  of  mass  M  at  rest,  the  two  parti- 
cles will,  after  inelastic  impact,  move 
on  together  with  the  common  veloc- 
ity v  =mu/(m -\- M).  Similarly  in 
our  case,  as  soon  as  the  impact 
has  taken  place,  the  two  masses  m 
and  M  may  be  regarded  as  forming 
a  whole,  and  as  moving  together. 
The  impulse  mu  acting  on  this  mass 
M+  m  at  O'  imparts  to  this  point 
a  certain  velocity  v  which  can  be 
determined.  As  the  particle  m  as- 


Fig.  53. 


sumes  this  velocity  v  after  impact,  it  loses  the  momentum  mu  —  mv, 
owing  to  the  impact ;  and  this  is  the  impulse 

=  m(u  —  v) 
of  the  blow  transmitted  to  the  body. 


206  KINETICS   OF   A  RIGID   BODY.  [369. 


To  find  v,  let  G'  be  the  centroid  of  M  +  m,  so  that 
GG'  =  mx}/(M+m),  G'O'  =  Mx^(M-\-  m).  The  principal  axes  of 
M+m  are  G'x  and  the  parallels  G'y',  G'z'  to  Gy,  Gz.  The  impulse 
mu  at  0'  imparts  to  the  mass  M+m  (see  Ex.  (4))  a  velocity  of 
translation  mu/(M-\-  m)  parallel  to  G'y',  and  an  angular  velocity 
<o  =  ;;/#•  G'O'/(M+  m)ql  about  6V,  ^  being  the  radius  of  inertia 
otM+m  for  6V.  The  velocity  v  of  (9'  is,  therefore, 

—      mu      i 
~~M+m 

For  q  we  have  the  relation 

m)f  =  Mq?  +  M-  GG'2+  m 


where  q^  is  the  radius  of  inertia  of  M  for  Gz.     Substituting  this  value  of 
q,  we  find 

v=mu 2* —  — z> 

and  hence 


)qf+  mx? 

It  thus  appears  that  F  equals  mu  only  in  the  limiting  case  when 
m  —  o,  u  =  oo,  whjle  lim  mu  =  const.  For  given  values  of  m  and  */,  F 
is  a  maximum  for  xl  =  o;  i.e.  when  w  strikes  the  body  M  at  the  cen- 
troid. In  this  case  f=  mMu/(Af+m),  as  it  should  be,  since  for 
direct  impact,  we  have 

F—  m(u  —  v)=mu  —  m-  mu/(M+  m)=  mu  •  M/(M+  m). 

(6)  A  free  rigid  body  turns  with  angular  velocity  o>  about  an  instan- 
taneous axis  1,  which  is  parallel  to  a  centroidal  principal  axis  and  meets 
another  centroidal  principal  axis  at  a  distance  GO  =  x  from  the  cen- 
troid G  (  Fig.  54) .  A  point  P  of  the  body,  situated  on  the  principal  axis 
GO  at  the  distance  GP=x //-<?#*  the  centroid,  strikes  a  fixed  obstacle; 
what  is  the  reaction  P  of  the  obstacle  9 

The  system  of  impulses  to  which  the  angular  velocity  o>  is  due  reduces 
to  an  impulse  F=  Mu  =  Mux  through  G,  at  right  angles  to  the  plane 
(/,  G),  and  a  couple  Fx^=  Ff  /x,  where  q  is  the  radius  of  inertia  for 
the  centroidal  axis  7  parallel  to  /.  The  vector  of  this  couple  is  parallel 
to  /  (see  Ex.  (4) ) .  Just  after  impact,  we  have,  in  addition,  the  im- 


370-] 


CONTINUOUS   MOTION    OF   FREE    BODY. 


207 


pulsive   reaction   P  of  the    fixed   point ;    hence  the  resultant  impulse 
=F-\-P,  the  resultant  couple  =  Fx^  +  Px. 

As  the   point  P  of  the  body  is  reduced  to   rest  by  the  impact,  we 
have  only  to  express  the  velocity  of 
P  and  equate  it  to  zero.     This  gives 
the  condition 


F+P 

M 


Mf 


x  =  o, 


whence 
p—  _ 


u 


since  xxi  =  g2.   This  becomes  =  —  F 
for  x  =  o  and  for  x  =  Xi. 

Show  that  there  are  two  points  of 
maximum  impact  on  GO  at  equal 
distances  from  /  on  opposite  sides,  and  that  the  maximum  impulse  is 


Fi     54 


(7)  A  free  rigid  body  turns  with  angular  velocity  o>  about  a  centroidal 
principal  axis  1  when  one  of  its  points  P,  situated  at  the  distance  x  from 
1  in   the  centroidal  plane  perpendicular  to   1,   strikes  a  fixed  obstacle. 
Determine  the  impulse  on   this  obstacle,   and  show  that  it  is  greatest 
when  x  =  q,  where  q  is  the  radius  of  inertia  for  I.  * 

(8)  In  Ex.  (6),  determine  the  initial  motion  of  the  body  after  striking 
the  fixed  obstacle. 

2.     CONTINUOUS    MOTION. 

370.  In  the  preceding  articles  (357-367)  it  has  been  shown 
how  to  determine  a  system  of  impulses  capable  of  producing 
any  given  instantaneous  state  of  motion  of  a  free  rigid  body. 
Any  change  in  the  state  of  motion  can  be  regarded  as  due  to 
a  system  of  forces  ;  and  by  reducing  the  effective  forces  of  the 
particles,  in  a  similar  way  as  has  been  done  for  the  momenta, 
this  system  of  forces  can  be  determined.  This  geometrical 
study  of  the  continuous  motion  produced  by  forces  is  here 
omitted,  as  it  would  require  a  more  complete  exposition  of  the 
theory  of  acceleration  than  has  been  given  in  the  first  part  of 
the  present  work. 


208  KINETICS    OF   A   RIGID    BODY.  [371. 

371.  Analytically,  the  continuous  motion  of  a  free  rigid  body 
is  given  by  the  six  equations  of  motion,  (4)  or  (5),  Art.  223,  and 
(6)  or  (7),  Art  224.     As  pointed  out  in  Art.  233,  the  motion  of 
the  centroid  and  the  motion  of  the  body  about  the  centroid  can 
be  considered  separately.     The  former  is  given  by  the  equations 
(8),  Art.  226,  viz.  : 

Mx  =  Rxy     MJ  =  Ry,     Mz=R«  (i) 

where  M  is  the  mass  of  the  body  ;  x,  j>,  z  are  the  components 
of  the  accelerations  of  the  centroid  along  any  three  fixed  rec- 
tangular axes  ;  and  Rm  Ry,  Rz  are  the  components  along  the 
same  axes  of  the  resultant  R  of  all  the  external  forces  acting 
on  the  body. 

The  motion  of  the  body  about  the  centroid  is  the  same  as  if 
the  centroid  were  fixed  (Art.  229).  It  is  therefore  best  studied 
by  taking  the  centroid  G  as  origin  ;  all  the  developments  of 
Arts.  324-356  will  then  apply  without  change,  except  that  the 
centroid  G  must  be  substituted  for  the  fixed  point  O.  The 
general  equations  (3),  Art.  326,  or  Euler's  equations  (4),  Art. 
328,  can  be  used  to  determine  the  motion  about  the  centroid. 

The  integration  of  Euler's  equations  gives  the  angular 
velocities  a>lt  <02,  <»3  about  the  three  centroidal  principal  axes  of 
the  body.  The  position  of  the  body,  i.e.  the  relation  of  this 
system  of  principal  axes  to  a  system  of  axes  through  the  cen- 
troid, parallel  to  a  fixed  system,  can  be  determined  by  means  of 
Euler's  angles  6,  <£,  ty  (see  Arts.  333-335),  or  by  means  of  the 
9  cosines  av  a2,  aB,  blt  b^  £3,  clt  c^  CB  (Arts.  336,  337). 

372.  Kinetic  Energy.     As  the  instantaneous  motion  consists 
of  an  angular  velocity  o>  about  the  instantaneous  axis  /  and  a 
velocity  of  translation  u  along  this  axis,  the  velocity  v  of  any 
point  of  the  body,  at  the  distance  r  from  /,  is  ?;  =  V«2+a>V2. 
Hence  the  kinetic  energy  (comp.  Art.  235)  has  the  expression 


o> 


)  = 


373-1  CONTINUOUS   MOTION   OF   FREE    BODY.  2OQ 

If  q,  q  denote  the  radii  of  inertia  of  the  body  for  the  instan- 
taneous axis  /  and  the  parallel  centroidal  axis  7,  we  have 


where  x  is  the  distance  between  /  and  L     Hence  denoting  by 
v  the  velocity  of  the  centroid,  v  =  V.z/2  +  aPx*,  we  find 

T=  \M(u*  +  o>2^2)  +  \  Mrfq*  =  \M&  +  \  M^co*.  (2) 

It  thus  appears  that  the  kinetic  energy  consists  of  two  parts, 
=  7\+  7*2,  one  of  which, 


may  be  called  the  kinetic  energy  of  the  centroid  (the  whole  mass 
M  being  regarded  as  concentrated  at  the  centroid),  while  the 
other, 


the  so-called  kinetic  energy  of  rotation,  is  the  kinetic  energy 
which  the  body  would  possess  if  it  were  rotating  with  the 
angular  velocity  o>,  not  about  the  instantaneous  axis  /,  but 
about  the  parallel  centroidal  axis  /  The  developments  of 
Arts.  322  and  323  apply  without  change  to  Tz. 

373.  Numerous  exercises  and  applications  will  be  found  in  the  works 
of  Price,  Besant,  Williamson  and  Tarleton,  Walton,  quoted  in  Art.  159  ; 
but  above  all  in  E.  J.  ROUTH,  Dynamics  of  a  system  of  rigid  bodies,  Ele- 
mentary part,  fifth  edition,  1891  ;  Advanced  part,  fourth  edition,  1884; 
London  and  New  York,  Macmillan.  Illustrations  and  examples,  as  well 
as  further  developments  of  the  theory,  will  also  be  found  in  the  works  of 
Schell  and  Budde  (Art.  159)  ;  in  the  French  collections  of  problems  by 
M.  Jullien  and  by  A.  de  Saint-Germain ;  in  J.  PETERSEN,  Lehrbuch  der 
Dynamik  fester  Korper,  deutsch  von  R.  von  Fischer-Benzon,  Kopeh- 
hagen,  Host,  1887;  E.  BOUR,  Cours  de  mecanique  et  machines,  IIP 
fascicule,  Paris,  Gauthier-Villars,  1874;  and  the  original  memoirs  of 
L.  POINSOT,  in  particular  his  Theorie  nouvelle  de  la  rotation  des  corps, 
Paris,  Bachelier,  1852  (also  in  Liouville's  Journal  de  mathematiques , 
Vol.  XVI.),  and  his  Precession  des  equinoxes,  Paris,  Mallet-Bachelier, 
1857.  Among  the  numerous  French  treatises  on  theoretical  mechanics 
those  of  Poisson,  Sturm,  Resal,  Collignon  deserve  especially  to  be 
mentioned  here. 

PART   III — 14 


2  T0  MOTION   OF   A  VARIABLE   SYSTEM.  [374. 


CHAPTER   VII. 

MOTION    OF    A    VARIABLE    SYSTEM. 

374.  In  the   present  chapter  we  shall   consider  very  briefly 
the  motion  of  a   general   system  of  n   particles,  connected   in 
any  way  whatever,  subject  to  any  conditions  or  constraints,  and 
acted  upon  by  any  forces. 

The  forces  can  be  distinguished  as  external  and  internal. 
The  latter  are  exerted  by  certain  particles,  or  groups  of  parti- 
cles, of  the  system  on  other  particles  of  the  same  system,  while 
the  former  proceed  from  without  the  system.  Thus,  in  con- 
sidering our  solar  system,  the  attractions  between  its  various 
members  are  internal  forces,  while  the  attractions  of  the  fixed 
stars  on  the  sun  or  planets  would  represent  external  forces. 

Besides  these  two  kinds  of  forces  there  may  be  forces  re- 
placing constraints,  such  as  reactions  of  fixed  points,  lines,  or 
surfaces,  friction,  etc. 

I.    Free  System. 

375.  If  the  system  be  free,  i.e.  if  it  be  only  acted  upon  by 
external    and   internal   forces,  while   there  are  no   constraints 
or  conditions  prescribed  for  it,  the  establishment  of  the  general 
equations  of  motion  is  simple,  although  their  integration  gen- 
erally presents   insuperable   difficulties.      The  problem  of  two 
bodies  (Arts.  150-158)  is  the  simplest  special  case. 

The  general  principles  laid  down  in  Arts.  218-238  for  the 
motion  of  a  rigid  body  apply  almost  without  change  to  a  free 
system  of  particles  ;  indeed,  they  apply  even  to  a  constrained 
system,  provided  that  all  conditions  and  constraints  are  replaced 


3770  FREE   SYSTEM.  211 

by  forces  and  that  these  constraining  forces  are  included  among 
the  forces  X,  Y,  Z,  acting  on  the  particles.  Thus  in  par- 
ticular, the  general  equations  of  motion  of  a  rigid  body,  viz. 
(4)  or  (5),  Art.  223,  and  (6)  or  (7),  Art.  224,  hold  for  a  variable 
system.  For  they  express  the  necessary,  though  not  in  general 
sufficient,  conditions  of  equilibrium  of  the  forces  acting  on  the 
particles  with  the  reversed  effective  forces  of  these  particles  ; 
and  this  equilibrium  is  not  changed  by  making  the  distances 
between  the  particles  invariable ;  i.e.  by  what  is  sometimes 
called  solidifying  the  system.  But  it  should  be  observed  that 
the  reductions  of  the  systems  of  momenta  and  effective  forces, 
given  in  the  chapter  on  the  rigid  body,  do  not  in  general  hold 
for  a  variable  system. 

376.  Let  F  be  the  resultant  of  all  the  external  and  internal 
forces  acting  on  one  of  the  n  particles ;  X,  Y,  Z  its  components 
along  a  system  of  fixed  rectangular  axes  ;  x,  y,  z  the  co-ordinates 
of  the  particle,  and  m  its  mass.     Just  as  in  Arts.  219,  220,  we 
have  the  equations  of  motion  of  the  particle 

mx=X,  my=-  Y,  mz  =  Z.  (i) 

There  are  3  such  equations  for  each  particle,  and  hence  3«  for 
the  whole  system.  These  $n  equations  express  the  equilibrium 
of  the  system  of  forces  composed  of  the  external,  internal,  and 
reversed  effective  forces. 

377.  Applying  the  principle  of  virtual  work  to  this  system 
of  forces,  we  find  d'Alemberfs  equation 

^(-mx  +  X}§x+-$(-my+  Y)ty  +  2(-mz  +  Z)Sz  =  o,  (2) 
in  which  Bx,  By,  82  are  the  components  of  an  arbitrary  displace- 
ment Bs  of  the  particle  m.  As  there  are  3/2  such  arbitrary 
component  displacements,  the  equation  (2)  is  equivalent  to  the 
3  72  equations  (i). 

If  written  in  the  form 

,  (3) 


212  MOTION   OF   A   VARIABLE    SYSTEM.  [378. 

it  expresses  the  equality  of  the  sum  of  the  virtual  works  of  the 
effective  forces  to  the  sum  of  the  virtual  works  of  the  external 
and  internal  forces,  for  any  infinitesimal  displacement  of  the 
system.  The  internal  forces  do  not  enter  into  this  equation  if 
they  occur  in  pairs  of  equal  and  opposite  forces,  as  will  usually 
be  the  case. 

378.  As  there  are  no  conditions,  we  may  select  for  Ss  the 
actual  displacement  ds  of  every  particle,  so  that  the  equation 
(3)  becomes 

^m(xdx+ydy+zdz)  =  ^(Xdx+  Ydy+Zdz}. 

The  left-hand  member  is  the  exact  differential  ^ 
=  ^^mz>2.      Hence,  integrating    between   the  limits  o  and 
and  denoting  by  VQ  the  velocity  of  the  particle  m  at  the  time  o 
we  find 

Ydy  +  Zdz).  (4 


This  is  the  equation  of  kinetic  energy.  The  right-hand  mem 
ber  represents  the  work  done  by  the  forces  during  the  time  /. 

379.  If  there  exists  a  force  function  or  potential  U  for  th< 
forces  X,  Yy  Z,  i.e.  if  these  forces  are  the  partial  derivative 
with  respect  to  x,  y,  z  of  one  and  the  same  function  U>  the  sys 
tern  is  said  to  be  conservative.  We  have  then 


and  the  integration  of  (4)  gives 


where  £70  is  the  value  of  U  for  t=o. 

Denoting  as  usual   the  kinetic  energy  by  T,  the  potential 
energy  —  U  by  V,  this  equation  can  be  written 

7—  T  4-  V  —  rnn <5t    •  (f\\ 

—  *  flT^   *  0 —  V»v**8ll«  ,  \\J 


it  expresses  the  principle  of  the  conservation  of  energy.     (Comp. 
Art.  79). 


382.]  FREE   SYSTEM. 


213 


380.  Exercises.     Show  the  existence  of  a  force-function  and  find  its 
expression  in  the  following  cases  (comp.  Art.  86)  : 

(1)  When  the  resulting  force  F  at  each  particle  m  is  constant  in 
magnitude  and  direction  (gravity). 

(2)  When  the  forces  F  are  all  attractions,  each  being  directed  to 
some  fixed  centre  O  and  a  function  of  the  distance  r  from  this  centre. 

(3)  When  the  forces  F  are  the  mutual  attractions  of  the  particles 
constituting  the  system. 

381.  A  variable  system  of  n  particles  possesses  a  centroid 
whose  co-ordinates  x,  y,  ~z  satisfy  the  equations 

M'X—  ^mx,  M  -  y  =  ^my,  M -~z  =  lanz. 

The  developments  of  Arts.  226,  227,  in  particular  the  principle 
of  the  conservation  of  linear  momentum,  or  the  principle  of  the 
conservation  of  the  motion  of  the  centroid,  hold  for  a  variable 
system  just  as  well  as  for  a  rigid  body.  The  position  of  the 
centroid  in  the  system  is  of  course  variable  with  the  time. 

The  principle  just  referred  to  asserts  that,  if  2X  =  o,  2Y  =  o, 
3Z  =  o,  the  centroid  of  the  system  is  at  rest,  or  moves  with  con- 
stant velocity  in  a  straight  line.  It  should  be  noticed  that  the 
conditions  2^=o,  2F=o,  ^Z=o  do  not  mean  that  there  are 
no  forces  at  all  acting  on  the  system ;  they  only  mean  that  the 
resultant  of  these  forces  reduces  to  zero  while  there  may  be  a 
resulting  couple  different  from  zero.  The  principle  would,  for 
instance,  apply  to  the  solar  system  if  the  action  of  the  fixed 
stars  be  regarded  as  vanishing  or  as  reducing  to  a  couple ;  the 
mutual  attractions  of  the  various  members  of  the  system  occur 
in  pairs  of  equal  and  opposite  forces,  and  have,  therefore,  a 
resultant  zero. 

382.  Similarly,  the  developments  of  Arts.  228-232,   in  par- 
ticular the  principle  of  the  conservation  of  angular  momentum, 
or  of  areas,  and  the  properties  of  the  invariable  plane,  apply 
without  change  to  the  free  variable  system. 


214  MOTION   OF   A   VARIABLE    SYSTEM,  [383. 

II.    System  Subject  to  Conditions. 

383.  The   constraints   and   conditions    to   which   a   variable 
system  is  subject  may  be  of  very  different  kinds.      In  general, 
however,  they  can  be  imagined  as  replaced  by  certain  forces, 
called  constraining-  forces  or  reactions,   by  the  introduction  of 
which  the  system  becomes  free.     On  the  other  hand,  it  may  be 
noticed  that  internal  forces,  such  as  tensions  of  connecting  rods 
or  strings,  can  sometimes  be  regarded  as  constraining  conditions. 

If  all  conditions  and  constraints  be  expressed  by  means  of 
forces,  and  these  forces  be  included  among  the  forces  X,  Y,  Z, 
the  equations  of  motion  of  the  particle  m  have  again  the  form 
(i),  Art.  376,  and  the  principle  of  virtual  work  gives  the  equa- 
tion of  d'Alembert  (2),  Art.  377.  But  it  should  be  noticed  that, 
in  general,  the  constraints  will  do  no  work  if  the  displacements 
&r,  By,  §z  are  properly  selected ;  in  other  words,  if  the  displace- 
ments be  taken  so  as  to  be  compatible  with  the  conditions  to 
which  the  system  is  subject,  the  constraining  forces  will  not 
enter  into  the  equation  (2).  This  is  d'Alembert's  principle. 

384.  Before  further  developing  this  idea  it  may  be  well  to 
indicate  here  the  considerations  by  which  d'Alembert  himself 
(and,  in  more  exact  language,  Poisson)  explained  his  celebrated 
principle. 

Any  particle  m  of  the  system  is  acted  upon  at  any  time  t 
by  two  kinds  of  forces,  the  given  external  and  internal  forces, 
whose  resultant  we  denote  by  F,  and  the  internal  reactions 

and  constraining  forces  whose 
resultant  we  call  F'  (Fig.  55). 
The  resultant  of  .Fand  F'  must 
be  geometrically  equal  to  the 
effective  force  mj,  where  j  is 
the  acceleration  of  the  particle 
at  the  time  t. 

Now,  if  we  introduce   at  m 
the  equal  and  opposite  forces  mj,  —mj,  the  action  of  ^and  F\ 


385.]  CONSTRAINED    SYSTEM.  21$ 

and  hence  the  motion  of  the  particle,  will  not  be  changed.  But 
we  can  now  combine  Fand  —mj'to  a  resultant  F".  Since  F,  F', 
—  7/z/are  in  equilibrium,  the  forces  .F'and  F"  are  in  equilibrium  ; 
/>.  F"  is  equal  and  opposite  to  F',  as  appears  from  the  figure. 

The  figure  also  shows  that  F  can  be  resolved  into  the  com- 
ponents mj  and  F" ;  the  former  component,  mj,  produces  the 
actual  motion  of  the  particle,  while  the  latter,  F",  is  consumed 
in  overcoming  the  internal  reactions  and  constraints  repre- 
sented by  F'.  This  component  F"  of  F  is  therefore  called  by 
d'Alembert  the  lost  force.  As  F'  +  F"  =  o  at  every  particle  of  the 
system,  d'Alembert's  principle  can  be  expressed  by  saying  that, 
at  every  moment  during  the  motion,  the  lost  forces  are  in  equilib- 
rium with  the  constraints  of  the  system. 

If  the  constraints,  instead  of  being  expressed  by  means  of 
forces,  are  given  by  equations  of  condition,  we  may  express 
the  same  idea  by  saying  that,  owing  to  the  given  conditions,  the 
.lost  forces  form  a  system  in  equilibrium. 

385.  We  shall  now  assume  that  the  constraints  or  conditions 
to  which  the  system  is  subject  are  expressed  by  means  of 
•equations  (the  case  of  conditions  expressed  by  inequalities  is 
•excluded)  between  the  co-ordinates  xt  y.  z  of  the  particles  and 
the  time  t.  In  the  most  general  case  these  equations  might 
also  contain  the  velocities  of  the  particles ;  this  case,  however, 
will  not  be  considered  here. 

Let  there  be  k  conditions 

<l>(t,  *i,  JV*i,  *2>  •"  )=o,     -^(t,  *i,  y\>  *i»  xv  '••  )>  —       (r) 

for  a  system  of  n  points.  Then  the  number  of  the  indepen- 
dent equations  of  motion  will  be  ^n  —  k.  For  these  equations 
must  express  the  equilibrium  of  the  given  forces,  together  with 
the  reversed  effective  forces,  under  the  given  conditions  ;  and 
for  this  equilibrium  it  is  sufficient  that  the  virtual  work  should 
-vanish  for  any  displacement  compatible  with  the  conditions,  the 
Avork  of  the  reactions  and  constraining  forces  being  zero  for 


2i6  MOTION    OF   A   VARIABLE    SYSTEM.  [386. 

such  virtual   displacement.     In    other   words,  in  d'Alembert's 
equation 

ot     (2) 


the  constraining  forces  due  to  the  conditions  will  not  appear  if 
the  displacements  &r,  fy,  82  be  so  selected  as  to  be  compatible 
with  the  k  conditions  (i). 

Now  this  will  be  the  case  if  these  displacements  be  made  to 
satisfy  the  equations  that  result  from  differentiating  the  condi- 
tions (i),  viz. 


=  o,  (3) 


It  should  be  noticed  that  in  this  differentiation,  or  rather 
variation,  the  time  t  is  regarded  as  constant.  If,  for  instance, 
one  of  the  conditions  (i)  constrain  a  particle  to  a  curve  or  sur- 
face varying  with  the  time,  say  the  surface  of  the  moving  earth, 
or  that  of  a  projectile  in  motion,  the  displacement  is  called 
virtual,  or  compatible  with  the  condition,  if  it  takes  place  on 
the  curve  or  surface  regarded  momentarily  as  fixed  (comp.  Art. 
193).  Indeed,  when  the  conditions  contain  the  time,  the  state- 
ment that  a  virtual  displacement  is  one  compatible  with  the 
conditions  has  no  definite  meaning ;  virtual  displacements  are 
then  defined  as  displacements  satisfying  the  equations  (3). 

386.  The  k  equations  (3)  make  it  possible  to  eliminate  k  of 
the  372  displacements  from  d'Alembert's  equation  (2).  There 
will  remain  ^n  —  k  independent  arbitrary  displacements,  whose 
coefficients  equated  to  zero  give  the  ^n—k  equations  of  motion. 

Applying  the  method  of  indeterminate  multipliers  (comp. 
Art.  194)  to  perform  this  elimination  in  a  systematic  way,  we 
have  to  multiply  the  k  equations  (i)  by  indeterminate  factors 
X,  /z,  •••,  and  to  add  them  to  equation  (2).  The  k  multipliers  X, 
//,,  •••  can  then  be  so  selected  as  to  make  the  coefficients  of  k  of 
the  3«  displacements  £r,  ty,  §z  vanish.  As  the  remaining  $11  — k 


388.] 


CONSTRAINED    SYSTEM. 


217 


displacements  are  arbitrary,  their  coefficients  must  also  vanish 
separately.  Thus  it  follows  that  the  coefficients  of  all  the 
displacements  in  the  resulting  equation  must  vanish,  and  we 
have  n  sets  of  3  equations  of  the  type 


my  = 


(4) 


r  *    •    »      i    •* 

It  is  apparent  from  these  equations  that  the   constraining 
force  acting  on  the  particle  m  has  the  components 


387.  It  has  thus  been   shown  that  a  system  of  n  particles 
subject  to   k   conditions    has  ^n  —  k  independent   equations  of 
motion.     The  equations  can  be  obtained  either  by  eliminating 
from  d'Alembert's  equation  (2)  k  of  the  3/2  displacements  £r, 
Sj>,  82  by  means  of  the  k  equations  (3),  and  then  equating  to 
zero  the  coefficients  of  the  remaining  3«  —  k  arbitrary  displace- 
ments, or  they  may  be  regarded  as  represented  by  the  3  n  equa- 
tions  (4),  since  these  equations   contain  k  arbitrary  quantities 
X,  fj,,   -•.     In  this  latter  form  they  are  sometimes  denoted  as 
Lagrange  s  first  form  of  the  equations  of  motion. 

388.  It  follows  from  the  remark  at  the  end  of  Art.  385,  that 
the   actual    displacements    dx,   dy,   dz  of  the  particles  can   be 
selected  as  virtual  displacements  only,   and  always,  when  the 
conditional  equations  (i)  do  not  contain  the  time.     If  this  con- 
dition be  fulfilled,   d'Alembert's    equation    (2)  can  be   written 

?w(xdr+ydy+~sidz)  =  l(Xdx+  Ydy  +  Zdz), 
or  d$kmiP  =  'St(Xdx+ Ydy  +  Zdz).  (6) 

This   relation    can    also  be  deduced   from  the  equations  (4) 
by  multiplying  them  by  xdt,  ydtt  zdt,  and  summing  the  equa- 


2l8  MOTION   OF   A   VARIABLE   SYSTEM.  [389. 

tions  for  all  the  particles.  The  left-hand  member  of  the  result- 
ing equation  is  again  dftymiP.  In  the  right-hand  member  we 
find,  besides  the  term  ^(Xdx+  Ydy  +  Zdz),  such  terms  as 


The  coefficients  of  X,  //,,  •••  vanish  only  when  the  conditions 
(i)  are  independent  of  the  time,  for  then  the  differentiation  of 
these  equations  (i)  gives 


In  other  words,  in  this  case  the  constraining  forces  do  no  work 
during  the  actual  displacement  of  the  system,  as  they  are  all 
perpendicular  to  the  paths  of  the  particles,  and  we  find  equa- 
tion (6). 

If,  however,  the  conditional  equations  (i)  contain  the  time, 
their  differentiation  gives 


and  we  find  in  the  place  of  equation  (6)  : 

Ydy  +  Zdz)  -\h-pfy  ••-.  (7) 


389.  If  the  conditional  equations  do  not  contain  the  time,  and 
if,  moreover,  there  exists  a  force-function  U  for  all  the  forces, 
•equation  (6)  can  be  put  into  the  form 


which  gives,  by  integration, 

^\m&  -  ^mv*  =  U-  £70,  (8) 

or,  by  putting  U=  —  V, 

r+F=7-0+F0.  (9) 

This    equation  expresses    the  principle  of  the  conservation  of 
energy. 

It  should  be  noticed  that,  even  when  there  exists  no  force- 
function,  the  elementary  work  ^(Xdx+  Ydy  +  Zdz)  is  a  quantity 
independent  of  the  co-ordinate  system,  and  the  sum  of  these 


390.]  CONSTRAINED   SYSTEM.  219 

•elementary  works  for  a  finite  time,  say  from  t=o  to  t=t,  repre- 
sents a  certain  finite  amount  of  work  W—  \  2  (Xdx+  Ydy  +  Zdz), 
.so  that  equation  (6)  gives  always 

^\mv*-^\mv*=  W. 

This  means  that  if  the  conditions  are  independent  of  the  time, 
the  increase  of  kinetic  energy  during  any  interval  of  time  is  equal 
to  the  work  done  during  this  time  by  all  the  external  and  internal 
forces. 

But  when  a  force-function  exists,  this  work  is  W=  U—  £70, 
where  U  is  a  function  of  the  co-ordinates  only.  The  work  done 
by  the  forces  depends  therefore  only  on  the  initial  and  final 
values  of  these  co-ordinates ;  i.e.  on  the  initial  and  final  con- 
figuration of  the  system,  but  not  on  the  character  of  the  motion 
by  which  the  system  is  brought  from  the  initial  to  the  final 
position. 

390.  It  has  been  shown  in  Art.  222  that,  for  an  invariable 
system  of  n  points,  i.e.  for  a  free  rigid  body,  the  number  of  condi- 
tions is  k  =  $n  —  6  ;  hence  the  number  of  independent  equations 
of  motions  of  a  free  rigid  body  is  372  —  ($n  —  6)  =6. 

A  rigid  body  with  a  fixed  axis  (Art.  291)  has  but  one  degree 
of  freedom  and  5  constraints ;  i.e.  its  position  is  given  by  a 
single  variable,  say  the  angle  of  rotation,  0,  about  the  fixed 
axis.  The  motion  of  such  a  body  is  therefore  given  by  a  single 
equation. 

A  rigid  body  that  can  turn  about  and  also  slide  along  a 
fixed  axis  has  4  constraints  and  2  degrees  of  freedom  ;  it  has 
therefore  2  equations  of  motion,  and  2  variables  are  sufficient  to 
determine  any  particular  position  of  the  body,  say  the  angle  6 
and  the  distance  x  measured  along  the  axis  of  rotation. 

A  rigid  body  with  one  fixed  point  (Art.  311)  is  an  example  of 
an  invariable  system  with  3  constraints  and  3  degrees  of  free- 
dom. Three  variables  are  necessary  and  sufficient  to  determine 
a  particular  position,  and  the  number  of  independent  equations 
of  motion  is  3. 


220  MOTION    OF   A   VARIABLE    SYSTEM.  [391. 

Similarly,  it  will  be  seen  in  every  other  case  that  a  rigid  body 
has  as  many  independent  equations  of  motion  as  it  has  degrees 
of  freedom,  or  as  it  requires  variables  to  fix  its  position. 
These  variables  may  be  called  the  co-ordinates  of  the  rigid  body. 
Thus  a  free  rigid  body  has  6  co-ordinates  corresponding  to  its 
6  degrees  of  freedom  and  6  equations  of  motion ;  we  might  take 
as  such  co-ordinates  the  co-ordinates  x,  j/,  ~z  of  the  centroid  and 
Euler's  angles  0,  (/>,  -v/r. 

391.  These  considerations  can  be  generalized  so  as  to  apply 
to  a  general  variable  system  of  n  points  with  k  conditions. 
Such  a  system  is  said  to  have  $n—k  =  m  co-ordinates  because 
it  has  $n—k  =  m  independent  equations  of  motion  (Art.  385). 
In  other  words,  in  the  place  of  the  3«  Cartesian  co-ordinates 
xy  y,  z  of  the  n  points,  subject  to  k  conditional  equations,  we 
may  introduce  $n  —  k=m  independent  variables,  say  qlt  q^ 
'"  <7m>  which  are  so  selected  as  to  satisfy  the  k  conditions  (i) 
identically.  These  variables  are  called  the  Lagrangian,  or 
generalized,  co-ordinates  of  the  system. 

By  the  introduction  of  these  new  variables  the  equations  of 
motion  (4)  assume  a  form  which  is  known  as  the  second  Lagran- 
gian form. 

Suppose,  for  instance,  that  the  system  is  subject  to  only  one 
condition,  viz.  that  one  point  Pl  of  the  system  should  remain  on 
the  surface  of  the  ellipsoid 


If  we  select  two  new  variables  q^  q^  connected  with  x^  yl9 
#!  by  the  equations  xl=acosql,  yl  =  b  sin  ql  cos  q^  z±  = 
*:  sin  ^  sin  02,  the  condition  </>  =  o  is  satisfied  identically  in  the 
new  co-ordinates  gv  02.  Hence,  by  introducing  qlt  q%  in 
the  place  of  xlt  j/1?  zlt  the  condition  <£  =  o  is  eliminated  from 
the  problem. 

We  now  proceed  to  establish  the  equations  of  motion  in  the 
second  Lagrangian  form,  for  a  variable  system  of  n  points-,  with 


392.]  CONSTRAINED    SYSTEM.  221 

the  k  conditions  (i),  i.e.  to  introduce  $n  —  k  =  m  new  variables 
or  generalized  co-ordinates,  glt  q^  •••  qm  in  the  place  of  the  3« 
Cartesian  co-ordinates  x^  yv  zv  x^  •••  zn,  selecting  the  new  co- 
ordinates so  as  to  satisfy  the  conditions  (i)  identically  (comp. 
Arts.  210-216). 

392.  The  Cartesian  co-ordinates  x,  y,  z  of  any  one  of  the  n 
points,  as  well  as  their  time  derivatives  x>  y,  z,  are  functions  of 
,<7i>  q^  •••  qm  and  of  the  time  t.  We  have  therefore 

dx  .  dx    •     ,  dx    •  ,    dx   • 


with  similar  expressions  for  y  and  z.  Thus  x,  y,  z  are  repre- 
sented as  functions  of  the  independent  variables  t,  q^  q^  •••  qm, 
<1\>  4  2?  '"  <?m- 

Differentiating  x  partially  with  respect  to  any  one,  q,  of  the 
•quantities  qlt  q^  •••  qm,  we  find 

dx=  &x        &x    .         &x    .  &x    . 

dq     dqdt     dqdq^          dqdq^ 


_d_dx_  ,J_dx_     -     ,  _d_  d^     .  d     dx      . 

dt  dq^dq^  dq  '  fl  '  3ft  dq  '  ***       ^  dqm  dq   ' 

We  have  therefore, 

dx_d^  dx_    ^_d_^2    ^.—  ——  (n\ 

l$q~~~dt  ^q     ^q~  '  dt  dq     dq~  dt  dq 

Again,  differentiating  (10)  partially  with  respect  to  q,  we  have 
dx__dx     dy^_dy     dz_fa  /     * 

TT  —  T~>      TT  —  T~">       T~T         ~     •  \*^/ 

dq     dq     dq     dq     dq     dq 
Let  us  also  form  the  derivatives  of  the  kinetic  energy 

),  (13) 


dT    ^    f-d^,'dy,-dz\ 
viz.  —-=2m  U^-+j//  +  ^— 

dq  \   dq       dq        dqj 

which  by  (11)  and  (12)  becomes 


dT          f.d  dx      .d  dy  .   -d  dz\  /T  ^ 

—  =  ^m(x--  —+y-r  -^  +  *^  TT  )  (14) 

dq  \  dt  dq       dt  dq       dt  dqj 


222  MOTION   OF   A   VARIABLE    SYSTEM.  [393. 

,  d  T     ^     (  .  dx  ,    •  dv  ,    •  dz\  .     . 

and  —=2m  \x—+y-^+z—   .  (15) 

dq  \   dq        dq       dqj 

From  (15)  and  (14)  we  find 


at   dq  \    dq        dq        dqj      dq 

393.  Thus  prepared  we  can  introduce  the  new  co-ordinates 
q  into  the  equations  of  motion  (4)  by  multiplying  these  equa- 
tions by  dx/dq,  dy/dq,  dz/dq,  and  adding  them  throughout  the 
whole  system  ;  this  gives  : 

•_,     f-.dx  ,   ••  dv  ,  ••  dz\      v>  /  vdx  ,    -,rdy  ,    ^dz\  /     * 

^m  (x  —  +y^+z—  ]  =  S(^r—  4-  F-^  +  Z—  ;          (17) 
\    dq     '  dq        dqj         \     dq          dq         dqj 

the  coefficients  of  X,  //,,  •••  all  disappear  in  the  summation,  since, 
by  hypothesis,  the  new  co-ordinates  satisfy  the  conditional 
equations  (i)  identically. 

The  right-hand  member  of  (17)  we  shall  denote  by  Q  (comp. 
Arts.  1  80,  211)  ;  the  left-hand  member  can  be  put  into  a  more 
convenient  form  by  means  of  (16)  and  (12).  Thus  we  find 
finally  the  equations  of  motion  in  the  second  Lagrangian  form  : 


at  dq      dq 

As  there  is  one  such  equation  for  every  one  of  the  Lagrangian 
co-ordinates  gv  q2,  •••  qm,  the  number  of  such  equations  is 
m  =  $n  —  k.  They  are  obtained  from  the  type  (18)  by  attaching 
successively  the  subscripts  i,  2,  •••  m  to  each  of  the  symbols 

q,  q,  Q- 

394.  In  the  particular  case  of  a  conservative  system,  i.e.  when 
there  exists  a  force  function  U  such  that 


—  —  ,  —  ——  ,  ,—  —    , 

vx  ay  az 

the  quantity  Q  in  (i  8)  is  evidently  =dU/dq)  so  that  the  equa- 
tions of  motion  assume  the  form 

dLsJl=^(T+U).  (19) 

dt  di      d    v 


39S-]  CONSTRAINED   SYSTEM.  223. 

This  equation  can  be  derived  more  directly  from  equation  (16) 
by  considering  an  infinitesimal  displacement  of  the  system.  If 
7  be  the  change  of  the  co-ordinate  q  in  such  a  displacement, 
the  partial  changes,  or  variations,  of  ;tr,  y,  z  will  be 


hence  the  work  of  the  effective  forces  m'x,  my,  mz,  for  the 
whole  system,  is 

,  "dy  ,  -dz\      5-  v     (..die  .  ..  dy  .  ..dz 
+y-^  +  z--}  =  §q^m(x—+y-/r  +  z— 
\   dq     '  dq        dqj  \    dq        dq        dq 

This  is  the  amount  by  which  the  potential  energy  V=  —  U  is 
diminished;  it  is,  therefore,  equal  to  (dU/dq}§q.  Hence  the 
first  term  in  the  right-hand  member  of  (16)  can  be  replaced 
by  dU/dq\  this  at  once  giyes  equation  (19). 

395.  From  Lagrange's  equations  it  is  easy  to  derive  Hamil- 
ton's principle. 

Let  each  of  the  equations  (18)  be  multiplied  by  the  infinitesi- 
mal displacement,  or  variation,  8$  ;  let  the  equations  be  added, 
multiplied  by  dt,  and  integrated  from  t±  to  t^  : 

a         (20) 

The  first  term  can  be  transformed  by  partial  integration  ;  remem- 
bering that  d(§q)/dt  =  §(dq)/dt,  we  have 

d  d 


s 
=  (  —  Sg)  —  I      — 

dt   dq    '  \dq    Vii     J**     $2 

If  now  the  variations  Sq  be  so  selected  as  to  vanish  both  at 
the  time  tl  and  at  the  time  /2,  the  first  term  vanishes  at  both 
limits.  Hence  equation  (20)  assumes  the  form 


As  s—  &g+—Sg  =  ST*nd  2g^  =  S^7for  a  conservative  sys- 
dq  dq 


224  MOTION   OF   A   VARIABLE    SYSTEM.  [396. 


tern,  and  =&W  for  a  general  system  (Art.  389),  the  equation 
reduces  to  the  simple  form 

o,  (21) 


in  the  general  case,  and  to 

=0,  or  Sj[V-  PV=o>  (22) 

in  the  case  of  a  conservative  system. 

396.  Hamilton's  principle  consists  in  the  proposition  that  the 
equation  (21)  or  (22)  holds  for  any  displacements  of  the  system 
compatible  with  the   conditions   (i),   provided   these   displace- 
ments be  zero  at  the  times  /x  and  t^.     Assuming  the  existence 
of  a  force-function,  i.e.  taking  (22)  as  the  expression  of  Hamil- 
ton's principle,  its  meaning  can  be  expressed  as  follows.     If  we 
consider  any  two  positions  of  the  moving  system,  say  the  posi- 
tions which  it  occupies  at  the  times  ^  and  /2,  the  motion  by 
which  the  system  actually  passes  from  the  former  to  the  latter 
position   is   characterized,    and   distinguished   from    any   other 
imaginable  ways  of  passing  from  the  former  to  the  latter,  by 
the  property  that  the  variation  of  the  time-integral  of  the  differ- 
ence between  kinetic  and  potential  energy  vanishes.     In  other 
words,  for  the  acttial  motion  the  average  value  of  the  difference 
between   kinetic  and  potential    energy  during  any   time   is  a 
minimum. 

The  chief  advantage  of  Hamilton's  principle  lies  in  the 
fact  that  it  is  independent  of  any  co-ordinate  system,  and  can 
therefore  be  used  as  a  convenient  starting-point  for  introducing 
the  variables  best  adapted  to  the  needs  of  the  particular  prob- 
lem. 

397.  A  more  complete  discussion  of  Lagrange's  equations  of  motion 
and  of  Hamilton's  as  well  as  other  similar  principles,  of  dynamics,  such 
as  the  principle  of  least  action,  of  least  constraint,  etc.,  will  be  found  in 
the  work  of  E.  J.  ROUTH  (see  Art.  373)  and  in  W.  SCHELL'S  Theorie  der 
Bewegung  und  der  Krafte,  Vol.  II.,  pp.  544-571.     The  kinetics  of  the 


397-]  CONSTRAINED   SYSTEM.  225 

variable  system  forms  the  basis  for  the  investigations  of  mathematical 
physics,  i.e.  for  the  theory  of  elastic  bodies,  of  fluid  and  liquid 
motion,  of  heat,  light,  electricity,  and  magnetism.  The  following 
works  are  particularly  recommended  as  introductions  to  this  subject : 
THOMSON  AND  TAIT,  Natural  Philosophy,  Vol.  L,  parts  i  and  2,  Cam- 
bridge, University  Press  (New  York,  Macmillan),  new  edition,  1890; 
W.  VOIGT,  Elementare  Mechanik  als  Einleitung  in  das  Studium  der 
•thforetischen  Physik,  Leipzig,  Veit,  1889 ;  G.  KIRCHHOFF,  Vorlesungen 
uber  mathematische  Physik ;  Mechanik,  3te  Auflage,  Leipzig,  Teubner, 
1883. 


PART   111 — 15 


ANSWERS. 


Pages  9,  10. 

(1)  40. 

(2)  (0)4*  and  5};  (b)  -  2$,  6f 

(3)  If    the   original  velocities   are   of  the   same   sense,   ^ 
v1  =  46^-  ;  if  not,  v  =  —  19^,  v1  =  lof  . 

(4)  e  =  o  gives  (a)  44^,    (b)   -  2^  ;   <?  =  i  gives  (0)  »  =  38^, 
v'  =  49i>  (£)»  =  -  55i>  »'  =  35f  • 

(5)  »  =  -«*.  (7)  SJft. 
(8)   (a)  0-31  ft;  W  9isec.;  (,)  66^  ft. 


(n)   («)  4  ft.  per  second;  (3)  28  ft.  per  second. 

(12)  For^  =  o:    (a)v  =  -  u\    (f}\\mv  —  Q\    (c)\imv=ur. 
w  -f-  m 


1H  —  Wl  i  2  W 

-  -  u,      v1  = 

m  -f  m'  m  + 

Hm  v1  =  o  ;  (V)  Hm  z;  =  2  #'  —  ^,  lim  v'  =  #'.     Interpret  these  results. 


T-I  /     \  —  i  /  z\      i* 

For  ^  =  i  :      (a)    v  —  -  -  u,      v1  =  -  -  u  ;      (b)    hm  v  =  —  u, 
m  -f  m'  m  +  m' 


Page  14. 

(1)  About  450  pounds;  9-375  and  0-191  foot-pounds. 

(2)  156-85  foot-poundals.  (6)  4^-  tons. 

(3)  363  foot-tons;  9  miles  per  hour.         (7)   13  and  2  foot-tons. 

Page  16. 
(i)  56-83  F.P.S.  units.  (2)   16  ft.  per  second. 

(3)  v  =  10,  /3  =  48F,  v'  =  i6i  ft'  =  isF- 

227 


228  ANSWERS. 

Pages  24,  25. 

(1)  (b)  250  pounds. 

(2)  (a)  8  ft.  per  second;   (b)  20  ft. 

(3)  (a)  825  pounds;   (b)  i\  miles;  (c)  about  1000  pounds. 

(4)  4-9  sec. 

(5)  It  would  be  greater  by  -^  oz. 

(6)  j  ==  (OT!  sin  0!  —  m2  sin  #2  —  /x^  cos  0!  —  /x2w2cos 
7"=  (sin  0j  -f  sin  02  —  /A!  cos  0j  +  /x,2  cos  02) niim.2g/(m^  + 

(7)  /=  5.4  ft.  per  second  ;  T=  f  pound. 

(9)  0-0363.  (n)   (0)  1528  pounds;  (b)  1910  pounds. 

(10)  0-025.  (I2)  5**9  ft- 

Pages  33,  34. 

(1)  (a)  15  270  foot-pounds;   (b)  30^  ft. 

(2)  (a)  917  pounds;  (b)  1557  pounds;  (c)  640  pounds. 

(3)  (a)  1267  foot-tons;   (b)  4435  foot-tons;  (c)  5  :  2. 

(4)  2016  foot-tons. 

(5)  864  ft.  per  second. 

(7)  («)  ^  = 

(8)  «*  = 

(9)  At  the  time  /,  let  s  be  the  distance  of  mi  and  m2;  slt  Ja 
their   distances    from   their    initial    positions,   so    that   sl  +  s  +  s2  =  s0. 


Then    we    find    s1=  —  ^  —  (s0-s),  s2=  —  ^  —  (SQ-S),    and 

^i  +  ^2 


f  —  )  =  2  K  (^!  -f  w2)  [-  —  -).      To  integrate,  put  j-  =  j-0  cos2  i<#>  ;   then 
\dt)  \s      sj 

we  find  /=-°\/  -  -  -  (<£  +  sin<f>).       The  particles  meet  at  the 
^  - 


-f 

2 


(ii)  (-)  .  W 


ANSWERS.  229 

(12)    L-^saA 


(14)  The  equation  —  =  —  p?x  —  2  K—    gives  :    (a)    when   KZ  >  /x2, 
dr  dt 

x  =  e~Kt  ( C1X't1-'*"  +  ^-^-^O,  from  which  it  can  be  shown  that 
the  particle  approaches  the  centre  asymptotically,  reaching  it  only  in 
an  infinite  time ;  if  Cl  and  C2  have  opposite  signs,  the  particle  will 
do  so  after  first  reaching  a  maximum  elongation  and  then  return- 
ing. (&)  When  K2  =  /x2,  x  =  e~Kt(Cl  +  Czt).  (c)  When  K2</x2,  x  = 
Ci<e~Kt  sin(V/x2  —  K2/+C2),  and  the  particle  performs  oscillations  of 
period  27T/V/A2  — *2  and  of  decreasing  amplitude  C^""'. 

Pages  36,  37. 

(1)  i  watt  =  0-73737   foot-pounds  per  second  =  0-001   341   H.P., 
i  H.P.  =  745  -9  watts. 

(2)  i  metric  H.P.  =  735-75  watts  =  0-9863  British  H.P. 

(3)  27!  H.P.  (6)  nearly  200  gallons. 

(4)  49iH-p-  (7)    35*54  gallons. 

(5)  (a)  64;   (J)  224384.  (8)   i  hour. 
(9)   (£)  88  hours;  (c)  about  21  weeks. 

Pages  46,  47. 

(2)  Taking  the  axis  of  z  vertically  upwards,  U=  UQ  —  mg(z  —  %) ; 
the  equipotential  surfaces  are  the  horizontal  planes  z  =  const. ;  the 
potential  energy  is  V  =  mg(z  —  ZQ)  . 

(4)  Taking  the  fixed  line  as  axis  of  z,  U——\f(r)dr\  the  equi- 
potential surfaces  are  circular  cylinders  about  the  axis  of  z. 


(5).  Let  r  =  V(x-xoy  +  (y-yoy  +  (z-zl))2  be  the  distance  of 
the  moving  point  (x,  y,  z)  from  the  fixed  centre  (x0,  y0,  z0)  ;  then  the 
direction  cosines  of  the  central  force  P=f(r)  are 


=          =,    ==, 

r  dx  r  dy  r  dz 

where  the  +  sign  holds  for  a  repulsion,  the  —  sign  for  an  attraction  ; 
hence    X=  ±f(r)  £,        F=  ±  /(r)  Z=  ±f(r)  |",  or,  putting 


X=±dF(r)/dx,   Y=±dF(r)/dy,   Z=± 
and  finally  U=  ±  F(r). 


230  ANSWERS. 

(6)    U= 


(9)   Compare  J.  C.  MAXWELL,  Electricity  and  magnetism,  Vol.  I., 
Arts.  118-123,  and  the  plates  I.-III. 

(10)  Taking  the  axis  of  z  vertically  upwards  and  denoting  by  r^  r% 
the  distances  from  Cl}  C2,  we  have  by  the  principle  of  kinetic  energy 
d(±mv2)  =  —  mgdz  —  Kmd*\/r?  +  K'mr.2dr2  ;  hence  the  equipotential  sur- 
faces are  (c  +  gz  —  ^K'r22)2rl2  =K2.  If  *'  =  o,  the  equation  becomes  for 
Ci  as  origin  (x*+y2  +  z2)  (c  +  gz)z  =  *2;  if  K  =  o,  we  find  for  C2  as 

origin  the  concentric  spheres  x2  +f  +  f  z  — 


Pages  52,  53. 

+  Ooj'o  —  J'o^o)  2=0. 


(2)  Fi  =  Kmmfi  has  the  ^-component  Xt  =  —  Kmm^  •  (x  —  xt)/rt 
=  —-Kmm{(x—xj)  ;  hence  2X=  —Km^.mj(x—x^,  '%¥=  —Km^n^y—y^, 
'%Z=  —  Km^m^z  —  Zf)  .  Equating  these  to  zero,  the  position  of  equili- 
brium is  found  as  the  centroid 


=  =  _ 

^m{  '  ^mt  '  mt 

of  the  masses  mt.  Taking  this  point  as  origin,  the  equation  of  the  plane 
of  motion  is  the  same  as  in  Ex.  (i)  ;  and  the  resultant  force  has  the 
components  —  Km^m{  -  x,  —Km^m^y,  —Km^m^z. 

Pages  65,  66. 

(2)  The  equation  of  the  orbit  given  in  Ex.  (i)  is  satisfied  not  only 
by  (x0,  y0)  ,  but  also  by  (X(]/K,  }'O/K)  ;  i.e.  the  orbit  passes  not  only 
through  the  initial  point  PQ,  but  also  through  the  point  Q,  which  is 
the  extremity  of  the  radius  vector  OQ—  VQ/K  parallel  to  VQ;  OPQ  and 
OQ  are  the  conjugate  semi-diameters  whose  equations  are  x0y=yox, 
xQy  =  yox. 

(4)  The  problem  reduces  to  that  of  constructing  the  axes  of  a  conic 
from  a  pair  of  conjugate  diameters. 

(8)  The  equation  of  a  central  conic  can  be  written  in  the  form 

---  +  -T—  r> 
/~  a*      P      aW 

where  /  is  the  perpendicular  from  the  centre  to  the  tangent  ;  the  upper 
sign  gives  the  ellipse,  the  lower  the  hyperbola.  Apply  (n),  Art.  IT  6. 


ANSWERS.  231 

(9)    (a)  Ellipse;  (b)  hyperbola;   (c)  parabola. 
(10)   Parabola  :  x— x^=  —  (y  —  y0)  —  ^L^L  {y  — j0)2,  where  2  a  is  the 

•distance  of  <93  from  the  point  O  that  bisects  <9i  O2 ;  the  point  midway 
between  O  and  O3  is  taken  as  origin,  and  OO3  as  axis  of  x. 


(n)  /ssItan-f 

/ 

Pages  76,  77. 


(4)  687  days. 

(5)  By  (24),  Art.  138,  v*  —  ^  -  T  -  ;  as  the  velocity  is  not  changed 

instantaneously,  we  must  have   —  —  T  -  =  —  —  T  —  ,  whence   the   new 

r       a        r        a' 

major  semi-axis  a'  can  be  found. 

(6)  An  ellipse  with  the  end  of  its  minor  axis  at  the  point  where  the 
change  takes  place. 

(7)  (a)  Ellipse  with  a  =  ^r\   (&)  parabola. 

(8)  Differentiate  (24),  Art.  138,  with  respect  to  /u,  and  a. 

(9)  The  periodic  time  T  would  be  diminished  by  —  T. 

m 


,     ^  i  ,.,.-. 

(10)   r  =  —  -      —  ;  hence  x  =  -  -,  y  =  -       —  ;  differ- 
i  +  e  cos  0  i  +  e  cos  0  i  +  e  cos  0 

entiating  and  remembering  that  rY/0/^=  c,  we  find 
dx          c    .    /i      dy      c 


•eliminating  0,  we  find  the  equation  of  the  hodograph 

_^'=,  or  since  ,  = 
(n)   1.016  914. 
(12)  /=JI? 


232  ANSWERS. 

Page  79. 

For.  —  ,, 


=  -.I;   for  n  =  -2,   f(r)  =  f(V  -  a2)  .  r  ;    for    »=i    and 

I/O 


=          ;  for  n  =  2  and  £  =  o,  f(r)  = 


(2)  Taking  the  diameter  perpendicular  to  the  force  as  axis  of  x> 
find  F-— 
initial  velocity. 


we  find  F-—  -  —  ,    where  0  is  the  radius,  <r  the  ^-component  of  the 


+  i  -  n 
(4) 


(5)  Ellipse,  parabola,  or  hyperbola,  according  as  /u,  —yfy*,  where 

j;0  is  the  initial  distance  from  the  plane,  y0  the  initial  velocity  perpen- 
dicular to  the  plane. 


Page  83. 

(2)  Let  p!,  p2  be  the  distances  of  mlt  m2  from  the  common  centroid 
at  any  time  /;  £i  =  Xi  —  x,  etc.;  then  the  equations  of  the  relative 
motion  are 

i    ^     &  </%  (m^rm,    \     ^2 

*  ^    '          ;       =   l7  "~  *    ' 


Pages  93,  94. 

(4)  (^)  7i  pounds;    (<5)  735.  6  pounds;    (<r)  1  20  ft.  per  sec. 

(5)  4840  pounds.  (9)  32-20  ft.  per  sec. 

(6)  e  =  1  —  inches.  (12)  4-7  pounds. 

4  » 

(7)  8°-6.  (13)   (a)  76permin.;  (b)  108. 
(14)  In  Fig.  23,  CD-.RF  =  PC\  PR,  hence  CD  =  ^-^  =  const, 


ANSWERS.  233: 

Pages  96,  97. 

(i)  The    integration    gives   tan  \  (ir  +  0)  =  tan  J  (TT  +00)  •  *    *  '  , 
which  gives  t  =  <x>  for  0  =  IT. 

(5)  The  particle  will  not  leave  the  circle  if  N  remains  positive; 

^=oif  0)50!  =  ---- 

3  I 

(6)  Greater  than  17-94  ft.  per  sec. 

(8)  The  tangential  force  is  mg  sin  0,  if  6  is  the  inclination  of  the 
tangent  to  the  horizon  ;  as  sin  0  =  dz/ds,  we  have 

dv        dz          j  d^s 

-*=**'  or  *^=**' 

whence  \  i?  —  ^  v$  =  g(z  —  z0)  .    The  result  also  follows  from  the  equa- 
tion of  kinetic  energy,  since  the  force-  function  is  C7=  mgz  -f  C. 

(9)  The  equation  of  motion  is  the  same  as  (8),  Art.  175,  except. 
that  g  is  replaced  by  g  sin  a. 

(10)  The  particle  performs  oscillations  of  period  2  TT//U,  if  F=  (Jr. 

Page  98. 

Taking  6  for  q,   we   find    Q  =  —  (mg  +  pnO)  K  ;    hence   the   inte- 


gration of  (17)  gives  tf  =  h  —  2  KgO  —  !-—02.    On  the  other  hand,  (14) 

m 


dt 


m 


Pages  102,  103. 

(3)  O)  9'8  in.;    (b)  1-23  oz. 

(4)  As  x  andjj'  are  small  of  the  first  order,  z  differs  from  r,  by  (5), 
only  by  a  small  quantity  of  the  second  order ;  hence  z  =  o,  z/r—  i,  so 
that  the  third  of  the  equations  (6)  gives  N=mg]  hence  x  =  —  gx/r, 
y  =  —  gy/r.     Integrating  and  putting  "*fgjr  —  /x,  we  find  x  =  C^  sin  /x/ 
4-  C2  cos  //,/,  y  =  Dl  sin  //,/  +  D%  cos  /x/.      Solving  for  sin  and  cos,  squar- 
ing and  adding,  we  find  an  ellipse  as  the  required  path,  just  as  in 
Art.  121. 


234 


ANSWERS. 


Page  112. 

(1)  The   differential   equation  is    r—  <oV  +  £"  sin  to/ =  o.     If  f=rQ> 
dr/dt  =  vQ  at  the  time  /=  o,  we  find 

(  g\«*     (  £\  -<*      8  • 

(2)  Let  0  be  the  angle,  at  the  time  /,  between  the  radius  CPt  drawn 
from  the  centre  C  of  the  circle  to  the  particle  P,  and  the  diameter  OCA 
through  the  fixed  point  O.     Then,  taking  O  as  origin,  and  the  initial 
position  OC$AQ  of  the  diameter  OCA  (when  P  is  at  A0)  as  axis  of  x, 
we  find 

dzO  .    „_.  .  fdO^ 


—.  +  o>2  sin  0  =  o,  whence    —    =  2  o>2  cos  0  +  C. 
<//  \"*/ 

As   the  absolute   initial   velocity  is   zero,   we   find   00  =  2  <«>.     Hence, 
<92  =  4  o>2  cos2  1  (9,  and  finally,  sin  1  ^  =  ^  ~  g_^,  or  tan  J  (TT  +  ^)  =  ^>'  . 

(3)   Let  xz+y*  =a2(i  +  «/")2  be  the  equation  of  the  circle,  whence 
-\-  y8y  =  o  ;   the  equation  of  motion  is  xBx  -f  j'S);  =  o  ;  eliminating 


£#,  S>>,  and  integrating,  we  find  xy  —yx  —  av^.     The  equation  of  the 
circle  gives  xx  +y'  =  aza(i  +  at)  ;  hence, 

a(i  +  a/)2*  =  ^«(i  +  a/)^  -  swV(i  +«02-^2- 
'To  integrate,  put  i  +  a/=  «r,  and  then  put  x  =  £r.     The  result  is 

x  =  a(i+at)  cos  —  ^  -  ,   y=  a(i  +  at)  sin 

'   * 


Pages  134,  135. 

The  square  of  the  radius  of  inertia  is  : 

(1)  K- 

(2)  OH/2;   (b)\tf;   WTV/2;   W 
(3)i/^  (8)  \a 

(4)  iV2-  (9)   i« 

(5)  2\«2-  (10)  i^ 

(6)  \h\  (n)  ^^ 

(7)  /2-  (12) 


ANSWERS.  235 


(13)  (a) 

(14)  I= 

Pages  137,  138. 

The  square  of  the  radius  of  inertia  is  : 


(2) 
(3) 

(4)  f  *2.  . 

(5)  i(^i2  +  ^22)  ;  in  the  limit,  /=  Ma2. 

(6)  Differentiating   the   moment   of  inertia  in   Ex.    (4),  we   find 
J=  M.\a\ 

(7)  (*H*2;  m*2;  Wi**  +  K- 

(9)  (*)K;    (6)±<*i    (c)^(a2  +  b2). 

(10) 


(12)  For  axis  parallel  to  b,  /=  J8[t(A3  +  ^)  +^(^  +  8)2]  ;  for  axis 
parallel  to  h,  I  —  f  8(1  /^3  4-  ^S2)  ;  for  perpendicular  axis, 
/=  £  8[/^3  +  ^  +  3  bh* 


Pages  149,  150. 

(i)  The  centroidal  principal  axes  are  perpendicular  to  the  faces. 
The  moments  for  these  axes  are  J-  M(&2+c9-),  1  M(c2+a2),  |  M(a?+P). 
The  central  ellipsoid  is  (£2  +  ^2)^+  (^  +  a2)/  4-  («2  +  <*V=  3  e4. 
For  an  edge  2  a,  f=$M(P+<*)  ;  for  a  diagonal  I= 


For  the  cube  the  central  ellipsoid  becomes  a  sphere  of  radius 
for  an  edge  of  the  cube,  /=  ^-  a5. 

(2)  Central  ellipsoid  :  (P  +  r>)  **  +  (^2  +  «2)  /  +  (<*  +  ^2)  ^2  =  5  e4  ; 
for/,^  =  i(6a2  +  ^2). 

(3)  Take  the  vertex  as  origin,  the  axis  of  the  cone  as  axis  of  x\ 
then  /j  =  -f^Ma2  ;  //,  i.e.  the  moment  of  inertia  for  the  jy£-plane,=|  Mh2. 
As  for  a  solid  of  revolution  about  the  axis  of  x  B*  =  C1  and  ^  =  (7,  we 


236  ANSWERS. 

have    /a'  =  /8'  =  $•/!,     and    /2  =  /3  =  /i'+i/i.        Hence,    72  =  73  = 
\  a2)  .     At  the  centroid  the  squares  of  the  principal  radii  are 


(4)  A  =  JS=C=lMa\    D  =  E  =  F=\Ma?;  hence  momental 
ellipsoid  :  4  (JT  +  f  -f  s2)  —  3  (jys  +  2*  +  ^)  =  6  ^  ;    squares  of  prin- 
cipal radii  :  \  a2,  {±  a2,  |i  a2. 

(5)  ?2  =  itf 

(6)  /= 


(8)  The  centroid  may  be  such  a  point  ;  or  if  the  central  ellipsoid  be 
an  oblate  spheroid,  the  two  points  on  the  axis  of  revolution  at  the  dis- 
tance ±  V(/i  —  I<?)IM  from  the  centroid. 

(9)  The  ellipsoid  must  have  the  same  central  ellipsoid  as  the  given 

body  ;   its  equation  is  —  -  +  *-.  -j  --  -  =  -5_  t  where  M  is  the  mass,  and 

A        Jj         G        JM 

A\  B',  C  are  the  moments  of  inertia  for  the  principal  planes  of  the 
body  at  the  centroid. 


\ 


Pages  163,  164. 


(2).fV2«.  (4) 

(5)  375  oo°  foot-pounds. 

(6)  —  -  -  •  —  ,  where  r  and  r*  are  expressed  in  feet. 
3600^    r1 

(7)  4  m.  22  s.  (8) 


Pages  203-207. 
>  =  F/Ma. 


(8)  The  body  begins  to  turn  about  an  axis  through  the  fixed  point, 
parallel  to  the  instantaneous  axis  before  impact,  with  angular  velocity 


, 

=^= 


928 


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